NBA Reproductive Analysis Project

Champions: 1991, 1992, 1993, 1996, 1997, 1998
Conference Titles: 1991, 1992, 1993, 1996, 1997, 1998
Division Titles: 1975, 1991, 1992, 1993, 1996, 1997, 1998, 2011, 2012

Overview

This assessment task allows you to consolidate and apply the concepts and skills you’ve learnt throughout the semester. This assessment requires you to generate a reproducible data analysis project.

Your reproducible data analysis project will be hosted as a repository on GitHub and you are required to submit the URL to your GitHub repository.

Scenario and Aim of the Data Analysis Project

You are a data analyst with the Chicago Bulls competing in the NBA (national basketball association). In the most recent NBA season (2018-19), your team placed 27th out of 30 (for win-loss record). Your team’s budget for player contracts next season is $118 million, ranked 26th out of 30 (for the purpose of this assignment, next season is 2019-20). For context, the team with the highest payroll budget is Portland with $148 million, while the best performing team was Milwaukee Bucks (who clinched the best league record in 2018-19 who clinched the best league record in 2018-29) with $131 million.

You have been tasked by the general manager of Chicago Bulls to find the best five starting players one from each position) your team can afford. (Make sure you don’t use up all of your money on just these five players, you still need to fill a full team roster, but are just focussed on finding five starting players here). You can choose players that are already playing for Chicago Bulls, you just need to prove that they are worth it.

Load required packages

# include all required packages at the start
library(tidyverse) 

## ── Attaching packages ──────────────────────────────────────────────────────────────────────── tidyverse 1.3.0 ──

## ✓ ggplot2 3.3.0     ✓ purrr   0.3.4
## ✓ tibble  3.0.1     ✓ dplyr   0.8.5
## ✓ tidyr   1.1.0     ✓ stringr 1.4.0
## ✓ readr   1.3.1     ✓ forcats 0.5.0

## ── Conflicts ─────────────────────────────────────────────────────────────────────────── tidyverse_conflicts() ──
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

library(prettydoc)
library(broom)
library(dplyr)
library(ggplot2)
library(knitr)
library(kableExtra)

## 
## Attaching package: 'kableExtra'

## The following object is masked from 'package:dplyr':
## 
##     group_rows

library(magrittr)

## 
## Attaching package: 'magrittr'

## The following object is masked from 'package:purrr':
## 
##     set_names

## The following object is masked from 'package:tidyr':
## 
##     extract

library(naniar)
library(stringi)
library(tidyr)
library(tinytex)
library(DT)

Data source

We have been provided the following data sets: 1. 2018-19_nba_player-statistics.csv : sourced from basketball-reference.com

2. 2018-19_nba_player-salaries.csv : sourced from hoopshype.com/salaries

3. 2019-20_nba_team-payroll.csv : sourced from hoopshype.com/salaries

4. 2018-19_nba_team-statistics_1.csv : sourced from basketball-reference.com

5. 2018-19_nba_team-statistics_2.csv : sourced from basketball-reference.com

Read Data

Read in the various files using the read_csv() function from the readr package.

p_stats <- read_csv("data/raw/2018-19_nba_player_statistics.csv")

## Parsed with column specification:
## cols(
##   .default = col_double(),
##   player_name = col_character(),
##   Pos = col_character(),
##   Tm = col_character()
## )

## See spec(...) for full column specifications.

p_sal <- read_csv("data/raw/2018-19_nba_player-salaries.csv")

## Warning: Missing column names filled in: 'X4' [4], 'X5' [5], 'X6' [6], 'X7' [7]

## Parsed with column specification:
## cols(
##   player_id = col_double(),
##   player_name = col_character(),
##   salary = col_double(),
##   X4 = col_logical(),
##   X5 = col_logical(),
##   X6 = col_logical(),
##   X7 = col_logical()
## )

team_stats <- read_csv("data/raw/2018-19_nba_team_statistics_1.csv")

## Warning: Missing column names filled in: 'X23' [23], 'X24' [24], 'X25' [25]

## Parsed with column specification:
## cols(
##   .default = col_double(),
##   Team = col_character(),
##   X23 = col_logical(),
##   X24 = col_logical(),
##   X25 = col_logical()
## )
## See spec(...) for full column specifications.

team_stats_2 <- read_csv("data/raw/2018-19_nba_team_statistics_2.csv")

## Parsed with column specification:
## cols(
##   .default = col_double(),
##   Team = col_character()
## )
## See spec(...) for full column specifications.

payroll <- read_csv("data/raw/2019-20_nba_team-payroll.csv")

## Parsed with column specification:
## cols(
##   team_id = col_double(),
##   team = col_character(),
##   salary = col_character()
## )

Tidying Data Process

Rename the variables.

#Rename the variables to remove % and variables starting with numbers
p_stats <- rename(p_stats,
                  FGp = 'FG%', x3P = '3P', x3PA = '3PA', x3Pp = '3P%', x2P = '2P', x2PA = '2PA', x2Pp = '2P%', eFGp = 'eFG%', FTp = 'FT%') 
team_stats <- rename(team_stats,
                     x3PAr = '3PAr', TSp = 'TS%', eFGp = 'eFG%', TOVp = 'TOV%', ORBp = 'ORB%', DRBp = 'DRB%')
team_stats_2 <- rename(team_stats_2,
                       FGp = 'FG%', x3P = '3P', x3PA = '3PA', x3Pp = '3P%', x2P = '2P', x2PA = '2PA', x2Pp = '2P%', FTp = 'FT%')

Dealing with NAs

#Replace the NAs found in shooting percentage of players who didn't attempt a particular shot
p_stats <- p_stats %>%
  mutate_if(is.numeric, funs(ifelse(is.na(.), 0, .)))

## Warning: funs() is soft deprecated as of dplyr 0.8.0
## Please use a list of either functions or lambdas: 
## 
##   # Simple named list: 
##   list(mean = mean, median = median)
## 
##   # Auto named with `tibble::lst()`: 
##   tibble::lst(mean, median)
## 
##   # Using lambdas
##   list(~ mean(., trim = .2), ~ median(., na.rm = TRUE))
## This warning is displayed once per session.

#Remove the NA columns added at the end of the Salary and Team Stats variables

p_sal <- p_sal[,-(4:7), drop = FALSE] # Removed columns 4 through to 7 that displayed no data
team_stats <- team_stats[, -(23:25), drop = FALSE] # Removed columns 23 through to 25 that displayed no data

Joining data frames

p_sal <- subset(p_sal, select = c(2:3)) # First remove player id as not required data

p_stats <- left_join(x = p_stats, y = p_sal) # This will join the salary to the respective Player

## Joining, by = "player_name"

p_stats <- drop_na(p_stats) # We can see that some Player salaries are missing, so we must remove these from our dataset to ensure we are 100% confident that our picks will keep us under the salary cap. 

which(is.na(p_stats), arr.ind = TRUE) # Quick test to identify if any NAs remain in the dataset

##      row col

Removing accents and periods

x <- stringi::stri_trans_general(p_stats$player_name, "Latin-ASCII")
 # Saves the player name list in new variable. Removes the accents and saves as a character vector

x_new <- as.data.frame(x, row.names = NULL, optional = FALSE, stringsAsFactors = FALSE) # Converts the vector into a single column data frame

x_new <- stringr::str_replace_all(x_new$x, pattern = "\\.", replacement = "")
# Removes the periodsin players names.

x_new <- as.data.frame(x, row.names = NULL, optional = FALSE, stringsAsFactors = FALSE) # Converts the vector into a single column data frame

x_new <- rename(x_new, player_name = 'x') # Changes column name to merge data frames

Combining Player Salary and Statistics

p_stats <- bind_cols(x = x_new, y = p_stats) # Combines the no accent name variable to the main data frame. 

p_stats <- subset(p_stats, select = -c(player_name1)) # Removes the player name variable that had the accents.

p_stats <- p_stats[, c(1,2,3,4,30,5:29)] # Move the salary variable into a more logical position in the table

p_stats <- rename(p_stats, Salary = 'salary') # Rename salary to Salary, to tidy up the appearance slightly.

Combining Team Statistics

comb_team <- merge(team_stats, team_stats_2, by.x = "Team", by.y = "Team") # Combine the two sheets, matching by the Team names as the order of the two sheets is different.

comb_team <- subset(comb_team, select = -c(2, 23)) # This will remove the 'Ranking' columns that appeared twice. They aren't necessary in this analysis, so have been removed.

comb_team <- comb_team[, c(1, 2, 22:23, 3:21, 24:44)] # The next three pieces of code, will reorganise the data into an order that is preferable for me. 

comb_team <- comb_team[, c(1:15, 36:44, 16:35)]

comb_team <- comb_team[, c(1:24, 33:44, 25:32)]

Removing of duplicate athletes, due to trades and filtering data

p_stats <- p_stats %>%
  group_by(player_name) %>%
  arrange(player_name, desc(G)) %>%
  distinct(player_name, .keep_all = TRUE) # Will remove any duplicate players based on the amount of games played in that row. The highest amount of games played for the duplicated remains. Team variable 'TOT' stands for Two or More Teams so that is the row we want to keep.

p_stats <- p_stats %>%
  filter(G >= 20, MP >= 100) # Filter out and remove players that haven't played enough and who's data could influence decisions unnecessarily. 

Transforming the Data

New variables

p_stats <- p_stats %>%
  group_by(Pos) %>%
  mutate(PTSpm = PTS / MP,
         FTpm = FT / MP,
         BLKpm = BLK / MP,
         ASTpm = AST / MP,
         STLpm = STL / MP,
         TOVpm = TOV / MP,
         x3Ppm = x3P / MP,
         PPG = PTS / G,
         APG = AST / G,
         RPG = TRB / G) # Creates new variables at the end of our data frame

p_stats <- arrange(p_stats, Pos) # Arranges the data frame in order of Position

p_stats <- p_stats %>%
  mutate_if(is.numeric, round, digits = 3)

## `mutate_if()` ignored the following grouping variables:
## Column `Pos`

datatable(p_stats, rownames = FALSE, filter="top", options = list(pageLength = 10, scrollX=T))

# Formatting the table with a scroll box so that it doesn't take up a considerable amount of the page. I have used the datatable function to include a search bar which will be helpful. 

write_csv(x = p_stats, path = "data/processed/p_stats.csv")
write_csv(x = comb_team, path = "data/processed/p_stats.csv") # Writing and saving the combined and new processed/tidy data. 

Explore various data models

Single Varible Linear Regression

Team Statistics

Pace v Wins

Pace_wins <- comb_team %>%
  ggplot(aes(Pace, W)) +
  geom_point(alpha = 0.5) +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Team Pace Ratings and Wins.

Pace_wins # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$Pace, y = comb_team$W, method = "pearson") # Checks the correlation between the two variables

## [1] 0.08892848

pace_wins_lm <- lm(W ~ Pace, data = comb_team)
summary(pace_wins_lm) # Creates a linear regression model for Pace v Wins.

## 
## Call:
## lm(formula = W ~ Pace, data = comb_team)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -23.7808  -7.1784   0.9702   8.7057  17.3618 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   -9.220    106.323  -0.087    0.932
## Pace           0.502      1.063   0.472    0.640
## 
## Residual standard error: 12.19 on 28 degrees of freedom
## Multiple R-squared:  0.007908,   Adjusted R-squared:  -0.02752 
## F-statistic: 0.2232 on 1 and 28 DF,  p-value: 0.6403

predict(pace_wins_lm) # Obtain the predicted/expected values.

##        1        2        3        4        5        6        7        8 
## 42.93945 40.78079 41.38320 40.32897 40.47958 39.27475 40.47958 39.82696 
##        9       10       11       12       13       14       15       16 
## 39.67636 41.43341 39.92736 40.02777 41.83502 42.58804 39.27475 40.07797 
##       17       18       19       20       21       22       23       24 
## 42.63824 41.08200 42.63824 40.78079 42.38723 40.02777 41.78482 41.23260 
##       25       26       27       28       29       30 
## 40.52978 42.53784 40.12817 41.08200 41.13220 41.68441

pace_wins_stdres <- rstandard(pace_wins_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
pace_wins_points <- 1:length(pace_wins_stdres) # Gives the length of the variable
pace_wins_labels <- if_else(abs(pace_wins_stdres) >= 1.5, paste(pace_wins_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = pace_wins_points, y = pace_wins_stdres)) +
  geom_point() +
  geom_text(aes(label = pace_wins_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified. 

pace_wins_hats <- hatvalues(pace_wins_lm) # Measures the leverage of the points. 

ggplot(data = NULL, aes(x = pace_wins_points, y = pace_wins_hats)) +
  geom_point()  # Shows the leverage on a graph

pace_wins_cook <- cooks.distance(pace_wins_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = pace_wins_points, y = pace_wins_cook))+
  geom_point() # shows the collective change through a scatterplot graph. 

car::durbinWatsonTest(pace_wins_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1       0.1024738      1.725845    0.46
##  Alternative hypothesis: rho != 0

pace_wins_res <- residuals(pace_wins_lm)
pace_wins_fitted <- predict(pace_wins_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values. 

ggplot(data = NULL, aes(x = pace_wins_fitted, y = pace_wins_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = pace_wins_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals.

Offence rating v Wins

of_wins <- comb_team %>%
  ggplot(aes(ORtg, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Team Offensive Ratings and Wins.

of_wins # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$ORtg, y = comb_team$W, method = "pearson") # Checks the correlation between the two variables

## [1] 0.8411322

of_wins_lm <- lm(W ~ ORtg, data = comb_team)
summary(of_wins_lm) # Creates a linear regression model for Offensive Rating v Wins.

## 
## Call:
## lm(formula = W ~ ORtg, data = comb_team)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -12.9184  -4.5433   0.0544   5.7203   8.6818 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -330.3378    45.1380  -7.318 5.73e-08 ***
## ORtg           3.3636     0.4087   8.230 5.88e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 6.621 on 28 degrees of freedom
## Multiple R-squared:  0.7075, Adjusted R-squared:  0.6971 
## F-statistic: 67.73 on 1 and 28 DF,  p-value: 5.882e-09

predict(of_wins_lm) # Obtain the predicted/expected values.

##        1        2        3        4        5        6        7        8 
## 33.26380 47.05442 38.30915 44.36357 22.16402 31.91837 37.63643 49.74527 
##        9       10       11       12       13       14       15       16 
## 36.29101 59.49962 58.15419 39.31822 47.72713 32.25473 26.53666 30.57294 
##       17       18       19       20       21       22       23       24 
## 52.43613 44.36357 44.36357 21.15495 40.66364 35.95465 48.39985 25.86395 
##       25       26       27       28       29       30 
## 55.46334 41.00000 49.40892 50.08163 42.68178 43.35450

of_wins_stdres <- rstandard(of_wins_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
of_wins_points <- 1:length(of_wins_stdres) # Gives the length of the variable
of_wins_labels <- if_else(abs(of_wins_stdres) >= 1.5, paste(of_wins_points), "")  # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = of_wins_points, y = of_wins_stdres)) +
  geom_point() +
  geom_text(aes(label = of_wins_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified. 

of_wins_hats <- hatvalues(of_wins_lm) # Measures the leverage of the points. 

ggplot(data = NULL, aes(x = of_wins_points, y = of_wins_hats)) +
  geom_point() # Shows the leverage on a graph

of_wins_cook <- cooks.distance(of_wins_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = of_wins_points, y = of_wins_cook))+
  geom_point()# shows the collective change through a scatterplot graph. 

car::durbinWatsonTest(of_wins_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1       0.1891721      1.501797    0.17
##  Alternative hypothesis: rho != 0

of_wins_res <- residuals(of_wins_lm)
of_wins_fitted <- predict(of_wins_lm)# Testing for homoscedasticity, and whether they have a constant variance across all x values. 

ggplot(data = NULL, aes(x = of_wins_fitted, y = of_wins_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = of_wins_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals

Defence rating v Wins

def_wins <- comb_team %>%
  ggplot(aes(DRtg, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Defensive Rating and Wins. Important to note that the negative linear relationship is actually reversed, as a higher defensive rating is not a good outcome for the team. A higher win rate and lower defensive rating is the ideal outcome.

def_wins # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$DRtg, y = comb_team$W, method = "pearson") # Checks the correlation

## [1] -0.7597828

def_wins_lm <- lm(W ~ DRtg, data = comb_team)
summary(def_wins_lm) # Creates a linear regression model for Defensive Rating v Wins

## 
## Call:
## lm(formula = W ~ DRtg, data = comb_team)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -13.7612  -6.0172  -0.6127   6.1833  13.1944 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 383.8923    55.4715   6.921 1.60e-07 ***
## DRtg         -3.1058     0.5023  -6.184 1.12e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.959 on 28 degrees of freedom
## Multiple R-squared:  0.5773, Adjusted R-squared:  0.5622 
## F-statistic: 38.24 on 1 and 28 DF,  p-value: 1.118e-06

predict(def_wins_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 30.14000 49.08547 43.18442 34.48814 32.31407 18.64849 40.07861 45.66907 
##        9       10       11       12       13       14       15       16 
## 44.73733 43.80559 40.07861 53.12303 37.59396 43.80559 45.97966 49.70663 
##       17       18       19       20       21       22       23       24 
## 57.16059 33.24582 34.17756 30.76117 51.57012 48.15373 42.25268 26.41303 
##       25       26       27       28       29       30 
## 40.69977 37.59396 38.52570 51.25954 55.60768 30.14000

def_wins_stdres <- rstandard(def_wins_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
def_wins_points <- 1:length(def_wins_stdres) # Gives the length of the variable
def_wins_labels <- if_else(abs(def_wins_stdres) >= 1.5, paste(def_wins_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = def_wins_points, y = def_wins_stdres)) +
  geom_point() +
  geom_text(aes(label = def_wins_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified. 

def_wins_hats <- hatvalues(def_wins_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = def_wins_points, y = def_wins_hats)) +
  geom_point() # Shows the leverage on a graph

def_wins_cook <- cooks.distance(def_wins_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = def_wins_points, y = def_wins_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(def_wins_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1     -0.06302277      2.123362   0.738
##  Alternative hypothesis: rho != 0

def_wins_res <- residuals(def_wins_lm)
def_wins_fitted <- predict(def_wins_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values. 

ggplot(data = NULL, aes(x = def_wins_fitted, y = def_wins_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = def_wins_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals.

3-Point Attempt Rating v Wins

x3PAr_wins <- comb_team %>%
  ggplot(aes(x3PAr, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between 3-Point Attempt Rate and Wins. 

x3PAr_wins # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$x3PAr, y = comb_team$W, method = "pearson") # checks the correlation

## [1] 0.2843269

x3PAr_wins_lm <- lm(W ~ x3PAr, data = comb_team)
summary(x3PAr_wins_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = W ~ x3PAr, data = comb_team)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -22.2461  -6.4940  -0.6239  11.4337  15.5671 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)    15.59      16.33   0.955    0.348
## x3PAr          70.82      45.13   1.569    0.128
## 
## Residual standard error: 11.74 on 28 degrees of freedom
## Multiple R-squared:  0.08084,    Adjusted R-squared:  0.04801 
## F-statistic: 2.463 on 1 and 28 DF,  p-value: 0.1278

predict(x3PAr_wins_lm) # Obtains the expected values

##        1        2        3        4        5        6        7        8 
## 44.13246 42.57449 44.13246 42.36204 36.48425 39.10447 45.47798 40.23754 
##        9       10       11       12       13       14       15       16 
## 43.49511 42.78694 52.34720 36.27180 36.48425 39.81264 39.81264 41.65387 
##       17       18       19       20       21       22       23       24 
## 45.26553 37.90059 38.53794 39.24611 40.16672 41.08734 39.81264 39.31692 
##       25       26       27       28       29       30 
## 39.60019 38.32549 35.84690 42.43286 43.49511 41.79551

x3PAr_wins_stdres <- rstandard(x3PAr_wins_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
x3PAr_wins_points <- 1:length(x3PAr_wins_stdres) # Gives the length of the variable
x3PAr_wins_labels <- if_else(abs(x3PAr_wins_stdres) >= 1.5, paste(x3PAr_wins_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = x3PAr_wins_points, y = x3PAr_wins_stdres)) +
  geom_point() +
  geom_text(aes(label = x3PAr_wins_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified. 

x3PAr_wins_hats <- hatvalues(x3PAr_wins_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = x3PAr_wins_points, y = x3PAr_wins_hats)) +
  geom_point()# Shows the leverage

x3PAr_wins_cook <- cooks.distance(x3PAr_wins_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = x3PAr_wins_points, y = x3PAr_wins_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(x3PAr_wins_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1     0.009106348      1.897535   0.772
##  Alternative hypothesis: rho != 0

x3PAr_wins_res <- residuals(x3PAr_wins_lm)
x3PAr_wins_fitted <- predict(x3PAr_wins_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = x3PAr_wins_fitted, y = x3PAr_wins_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = x3PAr_wins_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals.

True Shooting Percentage v Wins

TSP_wins <- comb_team %>%
  ggplot(aes(TSp, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between True Shooting Percentage and Wins. 

TSP_wins # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$TSp, y = comb_team$W, method = "pearson") # checks the correlation

## [1] 0.757103

TSP_wins_lm <- lm(W ~ TSp, data = comb_team)
summary(TSP_wins_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = W ~ TSp, data = comb_team)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -17.4251  -5.4489   0.4241   5.1456  16.8073 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -297.32      55.19  -5.387 9.63e-06 ***
## TSp           604.62      98.60   6.132 1.28e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.997 on 28 degrees of freedom
## Multiple R-squared:  0.5732, Adjusted R-squared:  0.558 
## F-statistic: 37.61 on 1 and 28 DF,  p-value: 1.283e-06

predict(TSP_wins_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 38.23891 45.49432 38.84353 37.63430 29.77427 29.16965 38.23891 40.05277 
##        9       10       11       12       13       14       15       16 
## 31.58812 63.02823 53.95897 41.86662 50.33126 37.63430 34.00659 30.37889 
##       17       18       19       20       21       22       23       24 
## 55.16820 36.42506 43.07585 22.51886 32.19274 35.21583 49.72665 36.42506 
##       25       26       27       28       29       30 
## 46.09894 37.63430 48.51741 52.74973 48.51741 45.49432

TSP_wins_stdres <- rstandard(TSP_wins_lm) # Gives the length of the variable
TSP_wins_points <- 1:length(TSP_wins_stdres) # Gives the length of the variable
TSP_wins_labels <- if_else(abs(TSP_wins_stdres) >= 1.5, paste(TSP_wins_points), "")# Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = TSP_wins_points, y = TSP_wins_stdres)) +
  geom_point() +
  geom_text(aes(label = TSP_wins_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed")# Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

TSP_wins_hats <- hatvalues(TSP_wins_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = TSP_wins_points, y = TSP_wins_hats)) +
  geom_point() # Shows the leverage

TSP_wins_cook <- cooks.distance(TSP_wins_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = TSP_wins_points, y = TSP_wins_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(TSP_wins_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1       0.0368267         1.777    0.54
##  Alternative hypothesis: rho != 0

TSP_wins_res <- residuals(TSP_wins_lm)
TSP_wins_fitted <- predict(TSP_wins_lm)# Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = TSP_wins_fitted, y = TSP_wins_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = TSP_wins_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Effective Field Goal Percentage

eFGp_wins <- comb_team %>%
  ggplot(aes(eFGp, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta")  # Testing for a linear relationship between Effective Field Goal Percentage and Wins. 

eFGp_wins # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$eFGp, y = comb_team$W, method = "pearson")# checks the correlation

## [1] 0.7818644

eFGp_wins_lm <- lm(W ~ eFGp, data = comb_team)
summary(eFGp_wins_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = W ~ eFGp, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -15.728  -5.514   1.954   4.208  14.272 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -282.36      48.75  -5.792 3.21e-06 ***
## eFGp          616.90      92.96   6.636 3.36e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.632 on 28 degrees of freedom
## Multiple R-squared:  0.6113, Adjusted R-squared:  0.5974 
## F-statistic: 44.04 on 1 and 28 DF,  p-value: 3.365e-07

predict(eFGp_wins_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 39.66339 47.06617 38.42959 34.72820 29.17611 27.94231 37.81269 42.74788 
##        9       10       11       12       13       14       15       16 
## 31.64370 66.19003 52.00136 44.59858 43.98168 42.74788 31.02680 35.34510 
##       17       18       19       20       21       22       23       24 
## 56.93655 32.87750 43.98168 19.92263 34.72820 37.19579 45.83237 34.72820 
##       25       26       27       28       29       30 
## 43.36478 40.89718 47.06617 52.61826 49.53377 45.21547

eFGp_wins_stdres <- rstandard(eFGp_wins_lm) # Gives the length of the variable
eFGp_wins_points <- 1:length(eFGp_wins_stdres) # Gives the length of the variable
eFGp_wins_labels <- if_else(abs(eFGp_wins_stdres) >= 1.5, paste(eFGp_wins_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = eFGp_wins_points, y = eFGp_wins_stdres)) +
  geom_point() +
  geom_text(aes(label = eFGp_wins_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

eFGp_wins_hats <- hatvalues(eFGp_wins_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = eFGp_wins_points, y = eFGp_wins_hats)) +
  geom_point() # Shows the leverage

eFGp_wins_cook <- cooks.distance(eFGp_wins_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = eFGp_wins_points, y = eFGp_wins_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(eFGp_wins_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1     -0.09648682      2.016169   0.966
##  Alternative hypothesis: rho != 0

eFGp_wins_res <- residuals(eFGp_wins_lm)
eFGp_wins_fitted <- predict(eFGp_wins_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = eFGp_wins_fitted, y = eFGp_wins_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = eFGp_wins_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Defensive Rebound Percentage v Lower Defensive Rating

defrb_drtg <- comb_team %>%
  ggplot(aes(DRBp, DRtg)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Defensive Rebound Percentage and the Defensive Rating. 

defrb_drtg # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$DRBp, y = comb_team$DRtg, method = "pearson") # checks the correlation

## [1] -0.5883396

defrb_drtg_lm <- lm(DRtg ~ DRBp, data = comb_team)
summary(defrb_drtg_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = DRtg ~ DRBp, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.7577 -1.5538  0.1272  1.0524  7.1279 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 186.0914    19.6640   9.464  3.2e-10 ***
## DRBp         -0.9821     0.2551  -3.850 0.000627 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.422 on 28 degrees of freedom
## Multiple R-squared:  0.3461, Adjusted R-squared:  0.3228 
## F-statistic: 14.82 on 1 and 28 DF,  p-value: 0.0006273

predict(defrb_drtg_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 111.0613 110.4721 111.0613 110.3739 110.1775 110.4721 109.9810 109.4900 
##        9       10       11       12       13       14       15       16 
## 108.8026 110.3739 113.0255 111.2577 111.4541 111.0613 109.8828 109.8828 
##       17       18       19       20       21       22       23       24 
## 107.2313 112.5344 110.6685 111.3559 109.2936 107.8205 108.9008 114.8914 
##       25       26       27       28       29       30 
## 109.5882 111.9452 108.1151 110.3739 107.2313 113.3201

defrb_drtg_stdres <- rstandard(defrb_drtg_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
defrb_drtg_points <- 1:length(defrb_drtg_stdres) # Gives the length of the variable
defrb_drtg_labels <- if_else(abs(defrb_drtg_stdres) >= 1.5, paste(defrb_drtg_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = defrb_drtg_points, y = defrb_drtg_stdres)) +
  geom_point() +
  geom_text(aes(label = defrb_drtg_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed")  # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

## Has an outlier at point number 6, need to explore this to see it's impact on the data. Doesn't appear to have a high leverage or high influence.

defrb_drtg_hats <- hatvalues(defrb_drtg_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = defrb_drtg_points, y = defrb_drtg_hats)) +
  geom_point() # Shows the leverage

defrb_drtg_cook <- cooks.distance(defrb_drtg_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = defrb_drtg_points, y = defrb_drtg_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(defrb_drtg_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1        0.233268      1.482338    0.16
##  Alternative hypothesis: rho != 0

defrb_drtg_res <- residuals(defrb_drtg_lm)
defrb_drtg_fitted <- predict(defrb_drtg_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = defrb_drtg_fitted, y = defrb_drtg_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = defrb_drtg_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals.

Turnover Totals contributing to Losses

tov_loss <- comb_team %>%
  ggplot(aes(TOV, L)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Turnover Totals and Losses

tov_loss # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$TOV, y = comb_team$L, method = "pearson") # checks the correlation

## [1] 0.2471353

tov_loss_lm <- lm(L ~ TOV, data = comb_team)
summary(tov_loss_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = L ~ TOV, data = comb_team)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -18.3705  -9.4752  -0.2251   7.5361  24.1344 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.16303   30.33643   0.005    0.996
## TOV          0.03536    0.02620   1.350    0.188
## 
## Residual standard error: 11.86 on 28 degrees of freedom
## Multiple R-squared:  0.06108,    Adjusted R-squared:  0.02754 
## F-statistic: 1.821 on 1 and 28 DF,  p-value: 0.188

predict(tov_loss_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 49.56487 37.36470 43.87146 35.56120 41.14852 39.27429 41.43143 39.13284 
##        9       10       11       12       13       14       15       16 
## 40.29982 41.50215 38.84994 39.84010 42.35086 45.56888 40.72417 42.88130 
##       17       18       19       20       21       22       23       24 
## 40.37054 38.14268 43.12884 40.86562 40.65344 38.42559 43.41174 45.39206 
##       25       26       27       28       29       30 
## 40.29982 38.88530 35.24293 40.83026 44.01291 40.97171

tov_loss_stdres <- rstandard(tov_loss_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
tov_loss_points <- 1:length(tov_loss_stdres) # Gives the length of the variable
tov_loss_labels <- if_else(abs(tov_loss_stdres) >= 1.5, paste(tov_loss_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = tov_loss_points, y = tov_loss_stdres)) +
  geom_point() +
  geom_text(aes(label = tov_loss_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

tov_loss_hats <- hatvalues(tov_loss_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = tov_loss_points, y = tov_loss_hats)) +
  geom_point() # Shows the leverage

tov_loss_cook <- cooks.distance(tov_loss_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = tov_loss_points, y = tov_loss_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(tov_loss_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1      0.09567978      1.784956   0.502
##  Alternative hypothesis: rho != 0

tov_loss_res <- residuals(tov_loss_lm)
tov_loss_fitted <- predict(tov_loss_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = tov_loss_fitted, y = tov_loss_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = tov_loss_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Net Rating v Wins

net_win <- comb_team %>%
  ggplot(aes(NRtg, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Team Net Rating and Wins

net_win # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$NRtg, y = comb_team$W, method = "pearson") # checks the correlation

## [1] 0.9800362

net_wins_lm <- lm(W ~ NRtg, data = comb_team)
summary(net_wins_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = W ~ NRtg, data = comb_team)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -5.099 -1.440  0.325  1.697  4.810 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 41.00808    0.44436   92.28   <2e-16 ***
## NRtg         2.42414    0.09294   26.08   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.434 on 28 degrees of freedom
## Multiple R-squared:  0.9605, Adjusted R-squared:  0.9591 
## F-statistic: 680.3 on 1 and 28 DF,  p-value: < 2.2e-16

predict(net_wins_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 26.94806 51.67431 40.76567 38.34152 20.64529 17.00907 37.85670 50.94706 
##        9       10       11       12       13       14       15       16 
## 40.52325 56.52259 52.64396 49.25016 43.18981 36.88704 34.46290 40.28084 
##       17       18       19       20       21       22       23       24 
## 61.85570 37.37187 38.09911 18.70597 49.00775 42.94739 47.31085 18.70597 
##       25       26       27       28       29       30 
## 51.18948 38.34152 45.12912 55.55293 53.61362 34.22048

net_wins_stdres <- rstandard(net_wins_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
net_wins_points <- 1:length(net_wins_stdres) # Gives the length of the variable
net_wins_labels <- if_else(abs(net_wins_stdres) >= 1.5, paste(net_wins_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = net_wins_points, y = net_wins_stdres)) +
  geom_point() +
  geom_text(aes(label = net_wins_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

net_wins_hats <- hatvalues(net_wins_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = net_wins_points, y = net_wins_hats)) +
  geom_point() # Shows the leverage

net_wins_cook <- cooks.distance(net_wins_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = net_wins_points, y = net_wins_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(net_wins_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1      -0.0192226      1.983334   0.952
##  Alternative hypothesis: rho != 0

net_wins_res <- residuals(net_wins_lm)
net_wins_fitted <- predict(net_wins_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = net_wins_fitted, y = net_wins_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = net_wins_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Total Points contributing to Offensive Rating

pts_off <- comb_team %>%
  ggplot(aes(PTS, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Points and Wins

pts_off # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$PTS, y = comb_team$W, method = "pearson") # checks the correlation

## [1] 0.6606001

pts_wins_lm <- lm(W ~ PTS, data = comb_team)
summary(pts_wins_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = W ~ PTS, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -16.215  -7.302   1.969   6.859  14.044 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.749e+02  4.641e+01  -3.770 0.000776 ***
## PTS          2.368e-02  5.086e-03   4.656  7.1e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.19 on 28 degrees of freedom
## Multiple R-squared:  0.4364, Adjusted R-squared:  0.4163 
## F-statistic: 21.68 on 1 and 28 DF,  p-value: 7.098e-05

predict(pts_wins_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 45.14174 43.29465 43.01048 40.09777 28.82580 27.92594 36.45096 39.95569 
##        9       10       11       12       13       14       15       16 
## 32.92255 53.57203 46.25473 34.79331 48.64647 42.08694 26.10253 30.31768 
##       17       18       19       20       21       22       23       24 
## 54.42453 43.46041 49.21480 28.11538 47.34403 33.44352 48.71751 33.79873 
##       25       26       27       28       29       30 
## 47.69924 46.77570 41.87381 47.27299 41.99222 46.46785

pts_wins_stdres <- rstandard(pts_wins_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
pts_wins_points <- 1:length(pts_wins_stdres) # Gives the length of the variable
pts_wins_labels <- if_else(abs(pts_wins_stdres) >= 1.5, paste(pts_wins_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = pts_wins_points, y = pts_wins_stdres)) +
  geom_point() +
  geom_text(aes(label = pts_wins_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

pts_wins_hats <- hatvalues(pts_wins_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = pts_wins_points, y = net_wins_hats)) +
  geom_point() # Shows the leverage

pts_wins_cook <- cooks.distance(pts_wins_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = pts_wins_points, y = pts_wins_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(pts_wins_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1       0.1627497      1.475814    0.15
##  Alternative hypothesis: rho != 0

pts_wins_res <- residuals(pts_wins_lm)
pts_wins_fitted <- predict(pts_wins_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = pts_wins_fitted, y = pts_wins_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = pts_wins_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Three points v Wins

x3_wins <- comb_team %>%
  ggplot(aes(x3P, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between successful 3 Point attempts and Wins

x3_wins # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$x3P, y = comb_team$W, method = "pearson") # checks the correlation

## [1] 0.4838773

x3_wins_lm <- lm(W ~ x3P, data = comb_team)
summary(x3_wins_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = W ~ x3P, data = comb_team)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -18.8669  -5.9449  -0.9138  10.2949  14.3599 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept) -2.94961   15.14840  -0.195  0.84702   
## x3P          0.04716    0.01612   2.926  0.00674 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.71 on 28 degrees of freedom
## Multiple R-squared:  0.2341, Adjusted R-squared:  0.2068 
## F-statistic:  8.56 on 1 and 28 DF,  p-value: 0.006744

predict(x3_wins_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 47.37509 45.72433 46.43180 43.13027 32.18807 36.99886 45.25268 39.64009 
##        9       10       11       12       13       14       15       16 
## 43.88491 48.31838 59.44925 33.79167 35.77258 36.99886 35.30094 40.81920 
##       17       18       19       20       21       22       23       24 
## 49.16735 36.05557 36.76304 35.86691 41.00786 41.24368 38.97978 34.31048 
##       25       26       27       28       29       30 
## 39.68725 40.77204 35.34810 44.92253 43.88491 40.91353

x3_wins_stdres <- rstandard(x3_wins_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
x3_wins_points <- 1:length(x3_wins_stdres) # Gives the length of the variable
x3_wins_labels <- if_else(abs(x3_wins_stdres) >= 1.5, paste(x3_wins_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = x3_wins_points, y = x3_wins_stdres)) +
  geom_point() +
  geom_text(aes(label = x3_wins_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

x3_wins_hats <- hatvalues(x3_wins_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = x3_wins_points, y = x3_wins_hats)) +
  geom_point() # Shows the leverage

x3_wins_cook <- cooks.distance(x3_wins_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = x3_wins_points, y = x3_wins_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(x3_wins_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1     -0.04675096       1.96371   0.882
##  Alternative hypothesis: rho != 0

x3_wins_res <- residuals(x3_wins_lm)
x3_wins_fitted <- predict(x3_wins_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = x3_wins_fitted, y = x3_wins_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = x3_wins_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Two points v Wins

x2_wins <- comb_team %>%
  ggplot(aes(x2P, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between successful Two Point Attempts and Wins

x2_wins # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$x2P, y = comb_team$W, method = "pearson") # checks the correlation

## [1] 0.06870566

x2_wins_lm <- lm(W ~ x2P, data = comb_team)
summary(x2_wins_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = W ~ x2P, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -23.416  -7.210   1.202   9.174  18.939 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) 29.701338  31.084806   0.955    0.348
## x2P          0.004636   0.012723   0.364    0.718
## 
## Residual standard error: 12.21 on 28 degrees of freedom
## Multiple R-squared:  0.00472,    Adjusted R-squared:  -0.03083 
## F-statistic: 0.1328 on 1 and 28 DF,  p-value: 0.7183

predict(x2_wins_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 40.48118 40.91701 40.15199 40.45799 41.38993 40.56000 39.71616 41.45948 
##        9       10       11       12       13       14       15       16 
## 39.86452 41.40847 38.48749 41.80721 41.58466 41.96022 40.37454 40.47190 
##       17       18       19       20       21       22       23       24 
## 41.06074 41.69130 42.40068 40.41627 41.59393 40.73155 41.37602 41.28793 
##       25       26       27       28       29       30 
## 41.59857 41.82112 42.01585 41.03756 40.46263 41.41311

x2_wins_stdres <- rstandard(x2_wins_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
x2_wins_points <- 1:length(x2_wins_stdres) # Gives the length of the variable
x2_wins_labels <- if_else(abs(x2_wins_stdres) >= 1.5, paste(x2_wins_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = x2_wins_points, y = x2_wins_stdres)) +
  geom_point() +
  geom_text(aes(label = x2_wins_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

x2_wins_hats <- hatvalues(x2_wins_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = x2_wins_points, y = x2_wins_hats)) +
  geom_point() # Shows the leverage

x2_wins_cook <- cooks.distance(x2_wins_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = x2_wins_points, y = x2_wins_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(x2_wins_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1        0.112963      1.721293    0.42
##  Alternative hypothesis: rho != 0

x2_wins_res <- residuals(x2_wins_lm)
x2_wins_fitted <- predict(x2_wins_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = x2_wins_fitted, y = x2_wins_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = x2_wins_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Free Throws v wins

ft_wins <- comb_team %>%
  ggplot(aes(FT, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Successful Free Throw Attempts and Wins 

ft_wins # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$FT, y = comb_team$W, method = "pearson") # checks the correlation

## [1] 0.1297686

ft_wins_lm <- lm(W ~ FT, data = comb_team)
summary(ft_wins_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = W ~ FT, data = comb_team)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -24.3841  -8.1118   0.3849   8.5255  18.7619 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept) 24.71592   23.61820   1.046    0.304
## FT           0.01123    0.01622   0.693    0.494
## 
## Residual standard error: 12.14 on 28 degrees of freedom
## Multiple R-squared:  0.01684,    Adjusted R-squared:  -0.01827 
## F-statistic: 0.4796 on 1 and 28 DF,  p-value: 0.4943

predict(ft_wins_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 40.92362 39.11528 42.18160 41.67616 39.63195 39.78920 42.02435 39.25006 
##        9       10       11       12       13       14       15       16 
## 40.60913 39.75550 42.48486 39.29499 45.52872 39.72180 41.03594 38.62107 
##       17       18       19       20       21       22       23       24 
## 41.23812 42.35008 41.13703 41.38413 41.12580 38.54245 44.28197 40.96855 
##       25       26       27       28       29       30 
## 42.21530 39.92398 40.53050 40.99101 42.01312 41.65370

ft_wins_stdres <- rstandard(ft_wins_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
ft_wins_points <- 1:length(ft_wins_stdres) # Gives the length of the variable
ft_wins_labels <- if_else(abs(ft_wins_stdres) >= 1.5, paste(ft_wins_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = ft_wins_points, y = ft_wins_stdres)) +
  geom_point() +
  geom_text(aes(label = ft_wins_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

ft_wins_hats <- hatvalues(ft_wins_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = ft_wins_points, y = ft_wins_hats)) +
  geom_point() # Shows the leverage

ft_wins_cook <- cooks.distance(ft_wins_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = ft_wins_points, y = ft_wins_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(ft_wins_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1       0.1088329       1.72528   0.492
##  Alternative hypothesis: rho != 0

ft_wins_res <- residuals(ft_wins_lm)
ft_wins_fitted <- predict(ft_wins_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = ft_wins_fitted, y = ft_wins_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = ft_wins_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Three Points v Offensive Rating

x3p_ortg <- comb_team %>%
  ggplot(aes(x3P, ORtg)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between successful 3 Point Attempts and Offensive Rating

x3p_ortg # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$x3P, y = comb_team$ORtg, method = "pearson") # checks the correlation

## [1] 0.5303433

x3p_ortg_lm <- lm(ORtg ~ x3P, data = comb_team)
summary(x3p_ortg_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = ORtg ~ x3P, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.4931 -2.1844  0.0529  2.0272  4.6598 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 98.354052   3.669782   26.80  < 2e-16 ***
## x3P          0.012927   0.003905    3.31  0.00257 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.595 on 28 degrees of freedom
## Multiple R-squared:  0.2813, Adjusted R-squared:  0.2556 
## F-statistic: 10.96 on 1 and 28 DF,  p-value: 0.002573

predict(x3p_ortg_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 112.1473 111.6949 111.8888 110.9839 107.9848 109.3033 111.5656 110.0273 
##        9       10       11       12       13       14       15       16 
## 111.1907 112.4059 115.4567 108.4243 108.9672 109.3033 108.8380 110.3504 
##       17       18       19       20       21       22       23       24 
## 112.6386 109.0448 109.2387 108.9931 110.4022 110.4668 109.8463 108.5665 
##       25       26       27       28       29       30 
## 110.0402 110.3375 108.8509 111.4751 111.1907 110.3763

x3p_ortg_stdres <- rstandard(x3p_ortg_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
x3p_ortg_points <- 1:length(x3p_ortg_stdres) # Gives the length of the variable
x3p_ortg_labels <- if_else(abs(x3p_ortg_stdres) >= 1.5, paste(x3p_ortg_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = x3p_ortg_points, y = x3p_ortg_stdres)) +
  geom_point() +
  geom_text(aes(label = x3p_ortg_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

x3p_ortg_hats <- hatvalues(x3p_ortg_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = x3p_ortg_points, y = x3p_ortg_hats)) +
  geom_point() # Shows the leverage

x3p_ortg_cook <- cooks.distance(x3p_ortg_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = x3p_ortg_points, y = x3p_ortg_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(x3p_ortg_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1      -0.1451083      2.200583   0.612
##  Alternative hypothesis: rho != 0

x3p_ortg_res <- residuals(x3p_ortg_lm)
x3p_ortg_fitted <- predict(x3p_ortg_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = x3p_ortg_fitted, y = x3p_ortg_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = x3p_ortg_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Two Points v Offensive Rating

x2p_ortg <- comb_team %>%
  ggplot(aes(x2P, ORtg)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between successful 2 Point Attempts and Offensive Rating

x2p_ortg # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$x2P, y = comb_team$ORtg, method = "pearson") # checks the correlation

## [1] 0.09895804

x2p_ortg_lm <- lm(ORtg ~ x2P, data = comb_team)
summary(x2p_ortg_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = ORtg ~ x2P, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.7404 -1.9357  0.0999  2.0059  6.0050 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1.063e+02  7.754e+00  13.714    6e-14 ***
## x2P         1.670e-03  3.174e-03   0.526    0.603    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.046 on 28 degrees of freedom
## Multiple R-squared:  0.009793,   Adjusted R-squared:  -0.02557 
## F-statistic: 0.2769 on 1 and 28 DF,  p-value: 0.6029

predict(x2p_ortg_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 110.2131 110.3701 110.0946 110.2048 110.5404 110.2415 109.9376 110.5655 
##        9       10       11       12       13       14       15       16 
## 109.9910 110.5471 109.4950 110.6907 110.6106 110.7459 110.1747 110.2098 
##       17       18       19       20       21       22       23       24 
## 110.4219 110.6490 110.9045 110.1897 110.6139 110.3033 110.5354 110.5037 
##       25       26       27       28       29       30 
## 110.6156 110.6958 110.7659 110.4135 110.2064 110.5488

x2p_ortg_stdres <- rstandard(x2p_ortg_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
x2p_ortg_points <- 1:length(x2p_ortg_stdres) # Gives the length of the variable
x2p_ortg_labels <- if_else(abs(x2p_ortg_stdres) >= 1.5, paste(x2p_ortg_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = x2p_ortg_points, y = x2p_ortg_stdres)) +
  geom_point() +
  geom_text(aes(label = x2p_ortg_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

x2p_ortg_hats <- hatvalues(x2p_ortg_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = x2p_ortg_points, y = x2p_ortg_hats)) +
  geom_point() # Shows the leverage

x2p_ortg_cook <- cooks.distance(x2p_ortg_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = x2p_ortg_points, y = x2p_ortg_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(x2p_ortg_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1       0.0257704      1.930104   0.876
##  Alternative hypothesis: rho != 0

x2p_ortg_res <- residuals(x2p_ortg_lm)
x2p_ortg_fitted <- predict(x2p_ortg_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = x2p_ortg_fitted, y = x2p_ortg_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = x2p_ortg_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Free Throws v Offensive Rating

ft_ortg <- comb_team %>%
  ggplot(aes(FT, ORtg)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between successful Free Throws and Offensive Rating

ft_ortg # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$FT, y = comb_team$ORtg, method = "pearson") # checks the correlation

## [1] 0.291237

ft_ortg_lm <- lm(ORtg ~ FT, data = comb_team)
summary(ft_ortg_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = ORtg ~ FT, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.1156 -1.7174  0.0868  2.2596  6.1985 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1.013e+02  5.698e+00  17.770   <2e-16 ***
## FT          6.304e-03  3.913e-03   1.611    0.118    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.929 on 28 degrees of freedom
## Multiple R-squared:  0.08482,    Adjusted R-squared:  0.05213 
## F-statistic: 2.595 on 1 and 28 DF,  p-value: 0.1184

predict(ft_ortg_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 110.3571 109.3422 111.0632 110.7795 109.6322 109.7205 110.9749 109.4179 
##        9       10       11       12       13       14       15       16 
## 110.1806 109.7015 111.2334 109.4431 112.9417 109.6826 110.4202 109.0649 
##       17       18       19       20       21       22       23       24 
## 110.5336 111.1577 110.4769 110.6156 110.4706 109.0207 112.2419 110.3823 
##       25       26       27       28       29       30 
## 111.0821 109.7961 110.1365 110.3950 110.9686 110.7669

ft_ortg_stdres <- rstandard(ft_ortg_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
ft_ortg_points <- 1:length(ft_ortg_stdres) # Gives the length of the variable
ft_ortg_labels <- if_else(abs(ft_ortg_stdres) >= 1.5, paste(ft_ortg_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = ft_ortg_points, y = ft_ortg_stdres)) +
  geom_point() +
  geom_text(aes(label = ft_ortg_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

ft_ortg_hats <- hatvalues(ft_ortg_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = ft_ortg_points, y = ft_ortg_hats)) +
  geom_point() # Shows the leverage

ft_ortg_cook <- cooks.distance(ft_ortg_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = ft_ortg_points, y = ft_ortg_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(ft_ortg_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1      0.04193079      1.894461   0.816
##  Alternative hypothesis: rho != 0

ft_ortg_res <- residuals(ft_ortg_lm)
ft_ortg_fitted <- predict(ft_ortg_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = ft_ortg_fitted, y = ft_ortg_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = ft_ortg_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Three Point Percentage v Wins

x3pp_w <- comb_team %>%
  ggplot(aes(x3Pp, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta")  # Testing for a linear relationship between 3 Point Shot Percentage and Wins

x3pp_w # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$x3Pp, y = comb_team$W, method = "pearson") # checks the correlation

## [1] 0.5418599

x3pp_w_lm <- lm(W ~ x3Pp, data = comb_team)
summary(x3pp_w_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = W ~ x3Pp, data = comb_team)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -21.7873  -5.8964   0.3398   7.7762  20.0635 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)  -110.23      44.37  -2.484  0.01923 * 
## x3Pp          425.41     124.70   3.411  0.00198 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.29 on 28 degrees of freedom
## Multiple R-squared:  0.2936, Adjusted R-squared:  0.2684 
## F-statistic: 11.64 on 1 and 28 DF,  p-value: 0.001983

predict(x3pp_w_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 39.51105 45.04142 39.93647 39.08564 39.08564 40.78729 34.40610 39.08564 
##        9       10       11       12       13       14       15       16 
## 37.80940 53.54969 41.21271 48.87014 54.82593 31.42820 35.25692 38.23481 
##       17       18       19       20       21       22       23       24 
## 39.93647 39.08564 36.10775 34.40610 37.80940 41.21271 42.48895 29.72655 
##       25       26       27       28       29       30 
## 42.48895 50.57180 56.52758 45.46684 41.21271 34.83151

x3pp_w_stdres <- rstandard(x3pp_w_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
x3pp_w_points <- 1:length(x3pp_w_stdres) # Gives the length of the variable
x3pp_w_labels <- if_else(abs(x3pp_w_stdres) >= 1.5, paste(x3pp_w_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = x3pp_w_points, y = x3pp_w_stdres)) +
  geom_point() +
  geom_text(aes(label = x3pp_w_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

x3pp_w_hats <- hatvalues(x3pp_w_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = x3pp_w_points, y = x3pp_w_hats)) +
  geom_point() # Shows the leverage

x3pp_w_cook <- cooks.distance(x3pp_w_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = x3pp_w_points, y = x3pp_w_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(x3pp_w_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1    -0.006206046      1.972432   0.942
##  Alternative hypothesis: rho != 0

x3pp_w_res <- residuals(x3pp_w_lm)
x3pp_w_fitted <- predict(x3pp_w_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = x3pp_w_fitted, y = x3pp_w_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = x3pp_w_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Three Point Percentage v Offensive Rating

x3pp_ortg <- comb_team %>%
  ggplot(aes(x3Pp, ORtg)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between 3 Point Shot Percentage and Offensive Rating

x3pp_ortg # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$x3Pp, y = comb_team$ORtg, method = "pearson") # checks the correlation

## [1] 0.5532274

x3pp_ortg_lm <- lm(ORtg ~ x3Pp, data = comb_team)
summary(x3pp_ortg_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = ORtg ~ x3Pp, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.1112 -1.8453  0.1448  1.7548  5.0457 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    71.79      11.00   6.528 4.48e-07 ***
## x3Pp          108.62      30.91   3.514  0.00152 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.55 on 28 degrees of freedom
## Multiple R-squared:  0.3061, Adjusted R-squared:  0.2813 
## F-statistic: 12.35 on 1 and 28 DF,  p-value: 0.001519

predict(x3pp_ortg_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 110.0198 111.4318 110.1285 109.9112 109.9112 110.3457 108.7165 109.9112 
##        9       10       11       12       13       14       15       16 
## 109.5854 113.6042 110.4543 112.4094 113.9300 107.9562 108.9337 109.6940 
##       17       18       19       20       21       22       23       24 
## 110.1285 109.9112 109.1509 108.7165 109.5854 110.4543 110.7802 107.5217 
##       25       26       27       28       29       30 
## 110.7802 112.8438 114.3645 111.5405 110.4543 108.8251

x3pp_ortg_stdres <- rstandard(x3pp_ortg_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
x3pp_ortg_points <- 1:length(x3pp_ortg_stdres) # Gives the length of the variable
x3pp_ortg_labels <- if_else(abs(x3pp_ortg_stdres) >= 1.5, paste(x3pp_ortg_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = x3pp_ortg_points, y = x3pp_ortg_stdres)) +
  geom_point() +
  geom_text(aes(label = x3pp_ortg_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

x3pp_ortg_hats <- hatvalues(x3pp_ortg_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = x3pp_ortg_points, y = x3pp_ortg_hats)) +
  geom_point() # Shows the leverage

x3pp_ortg_cook <- cooks.distance(x3pp_ortg_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = x3pp_ortg_points, y = x3pp_ortg_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(x3pp_ortg_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1      -0.1190102      2.189357   0.662
##  Alternative hypothesis: rho != 0

x3pp_ortg_res <- residuals(x3pp_ortg_lm)
x3pp_ortg_fitted <- predict(x3pp_ortg_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = x3pp_ortg_fitted, y = x3pp_ortg_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = x3pp_ortg_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Two Point Percentage v Wins

x2pp_w <- comb_team %>%
  ggplot(aes(x2Pp, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between 2 Point Shot Percentage and Wins

x2pp_w # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$x2Pp, y = comb_team$W, method = "pearson") # checks the correlation

## [1] 0.6009396

x2pp_w_lm <- lm(W ~ x2Pp, data = comb_team)
summary(x2pp_w_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = W ~ x2Pp, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -23.707  -9.320   2.787   7.149  11.738 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -145.27      46.86  -3.100 0.004375 ** 
## x2Pp          358.06      90.00   3.978 0.000445 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.785 on 28 degrees of freedom
## Multiple R-squared:  0.3611, Adjusted R-squared:  0.3383 
## F-statistic: 15.83 on 1 and 28 DF,  p-value: 0.0004453

predict(x2pp_w_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 40.20034 43.42285 38.41006 36.26172 32.32310 29.45865 42.34868 43.42285 
##        9       10       11       12       13       14       15       16 
## 33.75533 54.16454 52.01620 39.84228 36.26172 48.43564 35.54561 36.97783 
##       17       18       19       20       21       22       23       24 
## 57.02899 35.18755 46.64536 26.23614 37.33589 36.97783 44.13896 42.70674 
##       25       26       27       28       29       30 
## 41.99062 35.18755 38.41006 47.71953 48.43564 49.15175

x2pp_w_stdres <- rstandard(x2pp_w_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
x2pp_w_points <- 1:length(x2pp_w_stdres) # Gives the length of the variable
x2pp_w_labels <- if_else(abs(x2pp_w_stdres) >= 1.5, paste(x2pp_w_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = x2pp_w_points, y = x2pp_w_stdres)) +
  geom_point() +
  geom_text(aes(label = x2pp_w_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

x3pp_w_hats <- hatvalues(x3pp_w_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = x3pp_w_points, y = x3pp_w_hats)) +
  geom_point() # Shows the leverage

x3pp_w_cook <- cooks.distance(x3pp_w_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = x3pp_w_points, y = x3pp_w_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(x3pp_w_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1    -0.006206046      1.972432   0.904
##  Alternative hypothesis: rho != 0

x3pp_w_res <- residuals(x3pp_w_lm)
x3pp_w_fitted <- predict(x3pp_w_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = x3pp_w_fitted, y = x3pp_w_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = x3pp_w_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Two Point Percentage v Offensive Rating

x2pp_ortg <- comb_team %>%
  ggplot(aes(x2Pp, ORtg)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between 2 Point Shot Percentage and Offensive Rating 

x2pp_ortg # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$x3Pp, y = comb_team$ORtg, method = "pearson") # checks the correlation

## [1] 0.5532274

x2pp_ortg_lm <- lm(ORtg ~ x2Pp, data = comb_team)
summary(x2pp_ortg_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = ORtg ~ x2Pp, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.9585 -1.4956  0.1481  1.8522  4.0339 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    60.36      11.20   5.391 9.52e-06 ***
## x2Pp           96.19      21.50   4.473 0.000117 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.338 on 28 degrees of freedom
## Multiple R-squared:  0.4168, Adjusted R-squared:  0.3959 
## F-statistic: 20.01 on 1 and 28 DF,  p-value: 0.0001169

predict(x2pp_ortg_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 110.1852 111.0509 109.7042 109.1271 108.0690 107.2995 110.7623 111.0509 
##        9       10       11       12       13       14       15       16 
## 108.4538 113.9366 113.3594 110.0890 109.1271 112.3975 108.9347 109.3195 
##       17       18       19       20       21       22       23       24 
## 114.7061 108.8385 111.9166 106.4338 109.4157 109.3195 111.2433 110.8585 
##       25       26       27       28       29       30 
## 110.6661 108.8385 109.7042 112.2052 112.3975 112.5899

x2pp_ortg_stdres <- rstandard(x2pp_ortg_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
x2pp_ortg_points <- 1:length(x2pp_ortg_stdres) # Gives the length of the variable
x2pp_ortg_labels <- if_else(abs(x2pp_ortg_stdres) >= 1.5, paste(x2pp_ortg_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = x2pp_ortg_points, y = x2pp_ortg_stdres)) +
  geom_point() +
  geom_text(aes(label = x2pp_ortg_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

x2pp_ortg_hats <- hatvalues(x2pp_ortg_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = x2pp_ortg_points, y = x2pp_ortg_hats)) +
  geom_point() # Shows the leverage

x2pp_ortg_cook <- cooks.distance(x2pp_ortg_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = x2pp_ortg_points, y = x2pp_ortg_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(x2pp_ortg_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1      -0.1360498      2.229185   0.556
##  Alternative hypothesis: rho != 0

x2pp_ortg_res <- residuals(x2pp_ortg_lm)
x2pp_ortg_fitted <- predict(x2pp_ortg_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = x2pp_ortg_fitted, y = x2pp_ortg_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = x2pp_ortg_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Free Throw Percentage v Wins

ftp_w <- comb_team %>%
  ggplot(aes(FTp, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Free Throw Percentage and Wins 

ftp_w # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$FTp, y = comb_team$W, method = "pearson") # checks the correlation

## [1] 0.1611608

ftp_w_lm <- lm(W ~ FTp, data = comb_team)
summary(ftp_w_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = W ~ FTp, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -23.520  -7.355   1.762   9.610  18.637 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)    -5.18      53.49  -0.097    0.924
## FTp            60.21      69.68   0.864    0.395
## 
## Residual standard error: 12.08 on 28 degrees of freedom
## Multiple R-squared:  0.02597,    Adjusted R-squared:  -0.008814 
## F-statistic: 0.7466 on 1 and 28 DF,  p-value: 0.3949

predict(ftp_w_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 40.09884 43.10939 39.67737 42.80834 41.96538 42.50728 39.49673 40.27948 
##        9       10       11       12       13       14       15       16 
## 39.79779 43.04918 42.44707 40.09884 42.50728 36.90766 41.30306 36.66682 
##       17       18       19       20       21       22       23       24 
## 41.36327 42.20623 40.64074 40.52032 37.75061 41.90517 41.24285 41.72454 
##       25       26       27       28       29       30 
## 43.83192 38.53336 44.13298 43.22981 39.13547 41.06222

ftp_w_stdres <- rstandard(ftp_w_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
ftp_w_points <- 1:length(ftp_w_stdres) # Gives the length of the variable
ftp_w_labels <- if_else(abs(ftp_w_stdres) >= 1.5, paste(ftp_w_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = ftp_w_points, y = ftp_w_stdres)) +
  geom_point() +
  geom_text(aes(label = ftp_w_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

ftp_w_hats <- hatvalues(ftp_w_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = ftp_w_points, y = ftp_w_hats)) +
  geom_point() # Shows the leverage

ftp_w_cook <- cooks.distance(ftp_w_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = ftp_w_points, y = ftp_w_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(ftp_w_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1       0.1036315      1.742503   0.448
##  Alternative hypothesis: rho != 0

ftp_w_res <- residuals(ftp_w_lm)
ftp_w_fitted <- predict(ftp_w_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = ftp_w_fitted, y = ftp_w_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = ftp_w_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Free Throw Percentage v Offensive Rating

ft_ortg <- comb_team %>%
  ggplot(aes(FTp, ORtg)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Free Throw Percentage and Offensive Rating 

ft_ortg # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$FTp, y = comb_team$ORtg, method = "pearson") # checks the correlation

## [1] 0.4243103

ft_ortg_lm <- lm(ORtg ~ FTp, data = comb_team)
summary(ft_ortg_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = ORtg ~ FTp, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -6.2356 -0.5181  0.3085  1.7017  4.1508 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    80.00      12.27   6.518 4.59e-07 ***
## FTp            39.64      15.99   2.480   0.0194 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.772 on 28 degrees of freedom
## Multiple R-squared:   0.18,  Adjusted R-squared:  0.1508 
## F-statistic: 6.148 on 1 and 28 DF,  p-value: 0.01944

predict(ft_ortg_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 109.8067 111.7888 109.5292 111.5906 111.0356 111.3924 109.4103 109.9256 
##        9       10       11       12       13       14       15       16 
## 109.6085 111.7492 111.3527 109.8067 111.3924 107.7056 110.5995 107.5470 
##       17       18       19       20       21       22       23       24 
## 110.6392 111.1942 110.1635 110.0842 108.2606 110.9960 110.5599 110.8770 
##       25       26       27       28       29       30 
## 112.2645 108.7760 112.4627 111.8681 109.1724 110.4410

ft_ortg_stdres <- rstandard(ft_ortg_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
ft_ortg_points <- 1:length(ft_ortg_stdres) # Gives the length of the variable
ft_ortg_labels <- if_else(abs(ft_ortg_stdres) >= 1.5, paste(ft_ortg_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = ft_ortg_points, y = ft_ortg_stdres)) +
  geom_point() +
  geom_text(aes(label = ft_ortg_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

ft_ortg_hats <- hatvalues(ft_ortg_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = ft_ortg_points, y = ft_ortg_hats)) +
  geom_point() # Shows the leverage

ft_ortg_cook <- cooks.distance(ft_ortg_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = ft_ortg_points, y = ft_ortg_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(ft_ortg_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1     -0.01352301       2.01149    0.94
##  Alternative hypothesis: rho != 0

ft_ortg_res <- residuals(ft_ortg_lm)
ft_ortg_fitted <- predict(ft_ortg_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = ft_ortg_fitted, y = ft_ortg_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = ft_ortg_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Offensive Rebounds v Offensive Rating

orb_ortg <- comb_team %>%
  ggplot(aes(ORB, ORtg)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Offensive Rebounds and Offensive Rating 

orb_ortg # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$ORB, y = comb_team$ORtg, method = "pearson") # checks the correlation

## [1] 0.117081

orb_ortg_lm <- lm(ORtg ~ ORB, data = comb_team)
summary(orb_ortg_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = ORtg ~ ORB, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.9373 -2.3514  0.2509  2.0480  5.7247 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1.067e+02  5.964e+00  17.891   <2e-16 ***
## ORB         4.366e-03  6.998e-03   0.624    0.538    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.04 on 28 degrees of freedom
## Multiple R-squared:  0.01371,    Adjusted R-squared:  -0.02152 
## F-statistic: 0.3892 on 1 and 28 DF,  p-value: 0.5378

predict(orb_ortg_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 110.8651 110.2059 110.6250 110.2495 109.8304 110.5333 110.3281 110.9393 
##        9       10       11       12       13       14       15       16 
## 110.7822 110.1753 110.3456 110.0225 110.1709 110.3412 109.8522 110.7167 
##       17       18       19       20       21       22       23       24 
## 110.0225 110.7254 110.6643 110.4373 111.1969 110.2845 110.5901 109.9614 
##       25       26       27       28       29       30 
## 110.9175 110.6512 110.0007 110.1273 110.2757 110.1622

orb_ortg_stdres <- rstandard(orb_ortg_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
orb_ortg_points <- 1:length(orb_ortg_stdres) # Gives the length of the variable
orb_ortg_labels <- if_else(abs(orb_ortg_stdres) >= 1.5, paste(orb_ortg_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = orb_ortg_points, y = orb_ortg_stdres)) +
  geom_point() +
  geom_text(aes(label = orb_ortg_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

orb_ortg_hats <- hatvalues(orb_ortg_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = orb_ortg_points, y = orb_ortg_hats)) +
  geom_point() # Shows the leverage

orb_ortg_cook <- cooks.distance(orb_ortg_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = orb_ortg_points, y = orb_ortg_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(orb_ortg_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1      0.02811017      1.910839   0.794
##  Alternative hypothesis: rho != 0

orb_ortg_res <- residuals(orb_ortg_lm)
orb_ortg_fitted <- predict(orb_ortg_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = orb_ortg_fitted, y = orb_ortg_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = orb_ortg_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Defensive Rebounds v Defensive Rating

drb_drtg <- comb_team %>%
  ggplot(aes(DRB, DRtg)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Defensive Rebounds and Defensive Ratings

drb_drtg # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$DRB, y = comb_team$DRtg, method = "pearson") # checks the correlation

## [1] -0.5825676

drb_drtg_lm <- lm(DRtg ~ DRB, data = comb_team)
summary(drb_drtg_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = DRtg ~ DRB, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.9317 -2.4715  0.6419  1.4557  4.4717 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 143.355591   8.699642  16.478 6.08e-16 ***
## DRB          -0.011542   0.003043  -3.793  0.00073 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.434 on 28 degrees of freedom
## Multiple R-squared:  0.3394, Adjusted R-squared:  0.3158 
## F-statistic: 14.38 on 1 and 28 DF,  p-value: 0.0007305

predict(drb_drtg_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 110.7507 110.4737 109.6658 111.2932 111.0508 113.1283 110.0698 110.6699 
##        9       10       11       12       13       14       15       16 
## 111.5933 108.8464 113.1975 111.4317 109.4696 108.9041 112.1588 110.1275 
##       17       18       19       20       21       22       23       24 
## 105.0838 111.6163 109.0888 110.9123 109.7582 109.8620 108.4424 113.7746 
##       25       26       27       28       29       30 
## 109.1003 110.8200 109.7697 109.6543 108.9503 112.4358

drb_drtg_stdres <- rstandard(drb_drtg_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
drb_drtg_points <- 1:length(drb_drtg_stdres) # Gives the length of the variable
drb_drtg_labels <- if_else(abs(drb_drtg_stdres) >= 1.5, paste(drb_drtg_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = drb_drtg_points, y = drb_drtg_stdres)) +
  geom_point() +
  geom_text(aes(label = drb_drtg_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

drb_drtg_hats <- hatvalues(drb_drtg_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = drb_drtg_points, y = drb_drtg_hats)) +
  geom_point() # Shows the leverage

drb_drtg_cook <- cooks.distance(drb_drtg_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = drb_drtg_points, y = drb_drtg_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(drb_drtg_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1       0.1855995       1.55609   0.242
##  Alternative hypothesis: rho != 0

drb_drtg_res <- residuals(drb_drtg_lm)
drb_drtg_fitted <- predict(drb_drtg_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = drb_drtg_fitted, y = drb_drtg_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = drb_drtg_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Steals v Defensive Rating

stl_drtg <- comb_team %>%
  ggplot(aes(STL, DRtg)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Steals and Defensive Ratings

stl_drtg # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$STL, y = comb_team$DRtg, method = "pearson") # checks the correlation

## [1] -0.2123928

stl_drtg_lm <- lm(DRtg ~ STL, data = comb_team)
summary(stl_drtg_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = DRtg ~ STL, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.3024 -1.8397 -0.6048  1.9845  6.3658 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 116.058784   4.946220   23.46   <2e-16 ***
## STL          -0.009035   0.007855   -1.15     0.26    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.926 on 28 degrees of freedom
## Multiple R-squared:  0.04511,    Adjusted R-squared:  0.01101 
## F-statistic: 1.323 on 1 and 28 DF,  p-value: 0.2598

predict(stl_drtg_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 109.9603 109.6803 111.1891 110.7192 110.6108 111.2342 111.2433 110.3308 
##        9       10       11       12       13       14       15       16 
## 110.9180 110.4121 109.7345 109.6170 110.9903 110.4753 109.8790 110.3940 
##       17       18       19       20       21       22       23       24 
## 110.5024 109.8881 110.5476 111.0264 109.1382 111.1529 110.5837 109.4182 
##       25       26       27       28       29       30 
## 111.1258 109.9242 111.5324 109.9152 110.0687 109.8881

stl_drtg_stdres <- rstandard(stl_drtg_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
stl_drtg_points <- 1:length(stl_drtg_stdres) # Gives the length of the variable
stl_drtg_labels <- if_else(abs(stl_drtg_stdres) >= 1.5, paste(stl_drtg_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = stl_drtg_points, y = stl_drtg_stdres)) +
  geom_point() +
  geom_text(aes(label = stl_drtg_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

stl_drtg_hats <- hatvalues(stl_drtg_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = stl_drtg_points, y = stl_drtg_hats)) +
  geom_point() # Shows the leverage

stl_drtg_cook <- cooks.distance(stl_drtg_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = stl_drtg_points, y = stl_drtg_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(stl_drtg_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1      0.05713453      1.753875   0.522
##  Alternative hypothesis: rho != 0

stl_drtg_res <- residuals(stl_drtg_lm)
stl_drtg_fitted <- predict(stl_drtg_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = stl_drtg_fitted, y = stl_drtg_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = stl_drtg_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Steals v Offensive Rating

stl_ortg <- comb_team %>%
  ggplot(aes(STL, ORtg)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Steals and Offensive Rating

stl_ortg # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$STL, y = comb_team$ORtg, method = "pearson") # checks the correlation

## [1] 0.02283556

stl_ortg_lm <- lm(ORtg ~ STL, data = comb_team)
summary(stl_ortg_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = ORtg ~ STL, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.8315 -2.1159  0.2053  2.1810  5.5010 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1.098e+02  5.173e+00  21.222   <2e-16 ***
## STL         9.930e-04  8.216e-03   0.121    0.905    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.06 on 28 degrees of freedom
## Multiple R-squared:  0.0005215,  Adjusted R-squared:  -0.03517 
## F-statistic: 0.01461 on 1 and 28 DF,  p-value: 0.9047

predict(stl_ortg_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 110.4487 110.4795 110.3136 110.3653 110.3772 110.3087 110.3077 110.4080 
##        9       10       11       12       13       14       15       16 
## 110.3434 110.3990 110.4735 110.4864 110.3355 110.3921 110.4576 110.4010 
##       17       18       19       20       21       22       23       24 
## 110.3891 110.4566 110.3841 110.3315 110.5390 110.3176 110.3802 110.5083 
##       25       26       27       28       29       30 
## 110.3206 110.4527 110.2759 110.4537 110.4368 110.4566

stl_ortg_stdres <- rstandard(stl_ortg_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
stl_ortg_points <- 1:length(stl_ortg_stdres) # Gives the length of the variable
stl_ortg_labels <- if_else(abs(stl_ortg_stdres) >= 1.5, paste(stl_ortg_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = stl_ortg_points, y = stl_ortg_stdres)) +
  geom_point() +
  geom_text(aes(label = stl_ortg_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

stl_ortg_hats <- hatvalues(stl_ortg_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = stl_ortg_points, y = stl_ortg_hats)) +
  geom_point() # Shows the leverage

stl_ortg_cook <- cooks.distance(stl_ortg_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = stl_ortg_points, y = stl_ortg_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(stl_ortg_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1    -0.004529181      1.986446   0.942
##  Alternative hypothesis: rho != 0

stl_ortg_res <- residuals(stl_ortg_lm)
stl_ortg_fitted <- predict(stl_ortg_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = stl_ortg_fitted, y = stl_ortg_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = stl_ortg_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Assists v Offensive Rating

ast_ortg <- comb_team %>%
  ggplot(aes(AST, ORtg)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Assists and Offensive Rating

ast_ortg # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$AST, y = comb_team$ORtg, method = "pearson") # checks the correlation

## [1] 0.459058

ast_ortg_lm <- lm(ORtg ~ AST, data = comb_team)
summary(ast_ortg_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = ORtg ~ AST, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.0234 -1.9528 -0.1719  1.6681  7.3184 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 94.142343   5.966670  15.778 1.83e-15 ***
## AST          0.008064   0.002949   2.734   0.0107 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.72 on 28 degrees of freedom
## Multiple R-squared:  0.2107, Adjusted R-squared:  0.1825 
## F-statistic: 7.476 on 1 and 28 DF,  p-value: 0.01072

predict(ast_ortg_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 111.2217 111.5201 109.8992 109.5041 108.6251 107.8349 109.6089 112.2458 
##        9       10       11       12       13       14       15       16 
## 109.0203 113.6006 108.1816 111.3024 110.0283 111.0443 109.9718 110.1976 
##       17       18       19       20       21       22       23       24 
## 111.3669 110.4153 112.0120 107.4155 109.6009 111.0362 111.9394 109.9234 
##       25       26       27       28       29       30 
## 109.3589 110.9395 110.3750 110.9556 111.3427 111.5120

ast_ortg_stdres <- rstandard(ast_ortg_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
ast_ortg_points <- 1:length(ast_ortg_stdres) # Gives the length of the variable
ast_ortg_labels <- if_else(abs(ast_ortg_stdres) >= 1.5, paste(ast_ortg_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = ast_ortg_points, y = ast_ortg_stdres)) +
  geom_point() +
  geom_text(aes(label = ast_ortg_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed")  # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

# Point 11 has a std res value of just over 2.8, may be considered an outlier or a high leverage high influence point. 

ast_ortg_hats <- hatvalues(ast_ortg_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = ast_ortg_points, y = ast_ortg_hats)) +
  geom_point() # Shows the leverage

ast_ortg_cook <- cooks.distance(ast_ortg_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = ast_ortg_points, y = ast_ortg_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(ast_ortg_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1       -0.109389      2.170904   0.678
##  Alternative hypothesis: rho != 0

ast_ortg_res <- residuals(ast_ortg_lm)
ast_ortg_fitted <- predict(ast_ortg_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = ast_ortg_fitted, y = ast_ortg_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = ast_ortg_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Blocks v Defensive Rating

blk_drtg <- comb_team %>%
  ggplot(aes(BLK, DRtg)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "magenta") # Testing for a linear relationship between Blocks and Defensive Rating

blk_drtg # Prints the above graph

## `geom_smooth()` using formula 'y ~ x'

cor(x = comb_team$BLK, y = comb_team$DRtg, method = "pearson") # checks the correlation

## [1] -0.561718

blk_drtg_lm <- lm(DRtg ~ BLK, data = comb_team)
summary(blk_drtg_lm) # Creates a linear regression model.

## 
## Call:
## lm(formula = DRtg ~ BLK, data = comb_team)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.9633 -2.2885  0.1359  1.8858  5.0243 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 121.648806   3.162619  38.465  < 2e-16 ***
## BLK          -0.027687   0.007706  -3.593  0.00124 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.478 on 28 degrees of freedom
## Multiple R-squared:  0.3155, Adjusted R-squared:  0.2911 
## F-statistic: 12.91 on 1 and 28 DF,  p-value: 0.001238

predict(blk_drtg_lm) # Obtain the predicted/expected values

##        1        2        3        4        5        6        7        8 
## 110.0480 109.6050 112.2630 110.4356 111.9307 116.2499 111.9307 111.5985 
##        9       10       11       12       13       14       15       16 
## 112.4845 107.1132 110.4356 110.4633 110.9894 109.4666 109.2451 109.2451 
##       17       18       19       20       21       22       23       24 
## 108.1930 110.2695 109.4389 109.9650 109.8819 109.3282 109.6881 110.0757 
##       25       26       27       28       29       30 
## 110.2141 111.5985 110.9617 109.5497 108.2761 111.1555

blk_drtg_stdres <- rstandard(blk_drtg_lm) # Calculating the standardised residuals, which is the residual divided by their standard deviation
blk_drtg_points <- 1:length(blk_drtg_stdres) # Gives the length of the variable
blk_drtg_labels <- if_else(abs(blk_drtg_stdres) >= 1.5, paste(blk_drtg_points), "") # Will label the point on the graph if the residual is greater than 1.5 standard deviations.

ggplot(data = NULL, aes(x = blk_drtg_points, y = blk_drtg_stdres)) +
  geom_point() +
  geom_text(aes(label = blk_drtg_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "red", linetype = "dashed") # Create a graph with a straight line on the y-axis to show the limits where outliers are classified.

blk_drtg_hats <- hatvalues(blk_drtg_lm) # Measures the leverage of the points

ggplot(data = NULL, aes(x = blk_drtg_points, y = blk_drtg_hats)) +
  geom_point() # Shows the leverage

blk_drtg_cook <- cooks.distance(blk_drtg_lm) # Collective change in the coefficients when the ith point is deleted

ggplot(data = NULL, aes(x = blk_drtg_points, y = blk_drtg_cook))+
  geom_point() # Shows the collective change through a scatterplot graph.

car::durbinWatsonTest(blk_drtg_lm) # We begin to deal with the potential outliers by observing them. The Durbin Watson Test will determine if we have independence of observations. 

##  lag Autocorrelation D-W Statistic p-value
##    1       0.0515884      1.766672    0.44
##  Alternative hypothesis: rho != 0

blk_drtg_res <- residuals(blk_drtg_lm)
blk_drtg_fitted <- predict(blk_drtg_lm) # Testing for homoscedasticity, and whether they have a constant variance across all x values.

ggplot(data = NULL, aes(x = blk_drtg_fitted, y = blk_drtg_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") # Graphs the results of homoscedasticity

ggplot(data = NULL, aes(sample = blk_drtg_res)) +
  stat_qq() + stat_qq_line() # Normality of the residuals. 

Z-score for Team Points and breakdown of points

comb_team %>%
  mutate(z_pts = round((PTS - mean(PTS)) / sd(PTS)),
         pts_per_game = PTS / G,
         z_x3p = round((x3P - mean(x3P)) / sd(x3P))) %>%
  ggplot() +
  stat_qq(aes(sample = pts_per_game)) +
  facet_wrap(~ z_x3p) # The Z-Score is used to standardise and compare individual performances to the mean performances across a group. It allows us to adjust performances based on a given mean and standard deviation. 

# Plotting the z-score on a graph (stat_qq) tests of the normality to ensure there is no evidence of heteroscedasticity. 
  
comb_team %>%
  mutate(z_pts = round((PTS - mean(PTS)) / sd(PTS)),
         pts_per_game = PTS / G) %>%
  ggplot() +
  stat_qq(aes(sample = pts_per_game)) +
  facet_wrap(~ z_pts)  # The Z-Score is used to standardise and compare individual performances to the mean performances across a group. It allows us to adjust performances based on a given mean and standard deviation. 

# Plotting the z-score on a graph (stat_qq) tests of the normality to ensure there is no evidence of heteroscedasticity. 

Five Summary Statistics

comb_team %>%
  mutate(x3ppg = x3P / G,
         ptspg = PTS / G) %>%
  summarize(avg_3p = mean(x3ppg),
            s_3p = sd(x3ppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(x3ppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team 3 Point Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a team may make as well as the 3 point shots made.

##     avg_3p     s_3p avg_ptspg  s_ptspg         r
## 1 11.36382 1.504955  111.2085 4.092142 0.4565562

comb_team %>%
  mutate(x2ppg = x2P / G,
         ptspg = PTS / G) %>%
  summarize(avg_2p = mean(x2ppg),
            s_2p = sd(x2ppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(x2ppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team 2 Point Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a team may make as well as the 2 point shots made.

##     avg_2p     s_2p avg_ptspg  s_ptspg         r
## 1 29.71829 2.173743  111.2085 4.092142 0.3105844

comb_team %>%
  mutate(ftppg = FT / G,
         ptspg = PTS / G) %>%
  summarize(avg_ft = mean(ftppg),
            s_ft = sd(ftppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(ftppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Free Throw Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a team may make as well as the Free Throw shots made.

##     avg_ft     s_ft avg_ptspg  s_ptspg         r
## 1 17.68049 1.694804  111.2085 4.092142 0.4015758

comb_team %>%
  mutate(astpg = AST / G,
         ptspg = PTS / G) %>%
  summarize(avg_ast = mean(astpg),
            s_ast = sd(astpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(astpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Assists Per Game and Points Per Game. It gives us the ability to predict the amount of points a team may make as well as the Assists made.

##    avg_ast   s_ast avg_ptspg  s_ptspg         r
## 1 24.58659 2.08829  111.2085 4.092142 0.5653053

comb_team %>%
  mutate(stlpg = STL / G,
         ptspg = PTS / G) %>%
  summarize(avg_stl = mean(stlpg),
            s_stl = sd(stlpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(stlpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Steals Per Game and Points Per Game. It gives us the ability to predict the amount of points a team may make as well as the Steals made.

##   avg_stl     s_stl avg_ptspg  s_ptspg        r
## 1 7.63374 0.8436119  111.2085 4.092142 0.171338

comb_team %>%
  mutate(orbpg = ORB / G,
         ptspg = PTS / G) %>%
  summarize(avg_orb = mean(orbpg),
            s_orb = sd(orbpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(orbpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Offensive Rebounds Per Game and Points Per Game. It gives us the ability to predict the amount of points a team may make as well as the Offensive Rebounds made.

##    avg_orb     s_orb avg_ptspg  s_ptspg         r
## 1 10.34715 0.9837691  111.2085 4.092142 0.1915307

comb_team %>%
  mutate(drbpg = DRB / G,
         ptspg = PTS / G) %>%
  summarize(avg_drb = mean(drbpg),
            s_drb = sd(drbpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(drbpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Defensive Rebounds Per Game and Points Per Game. It gives us the ability to predict the amount of points a team may make as well as the Defensive Rebounds made.

##    avg_drb    s_drb avg_ptspg  s_ptspg         r
## 1 34.81829 1.811346  111.2085 4.092142 0.5642984

Position Specific Data Frames

New Data Frames for Specific Position

centres <- subset(p_stats, Pos == "C") # Creates a data frame specific for Centres

sg <- subset(p_stats, Pos == "SG") # Creates a data frame specific for Shooting Guards

sf <- subset(p_stats, Pos == "SF") # Creates a data frame specific for Small Forwards

pf <- subset(p_stats, Pos == "PF") # Creates a data frame specific for Power Forwards

pg <- subset(p_stats, Pos == "PG") # Creates a data frame specific for Point Guards

write_csv(x = pg, path = "data/processed/pg.csv") # Saves the new PG data into a new computer file.

write_csv(x = sg, path = "data/processed/sg.csv") # Saves the new SG data into a new computer file.

write_csv(x = sf, path = "data/processed/sf.csv") # Saves the new SF data into a new computer file.

write_csv(x = pf, path = "data/processed/pf.csv") # Saves the new PF data into a new computer file.

write_csv(x = centres, path = "data/processed/centres.csv") # Saves the new C data into a new computer file.

Linear Regression

Point Guard

reg_pg_drb_ppg <- pg %>%
  mutate(drbpg = DRB / G,
         ptspg = PTS / G) %>%
  ggplot(aes(drbpg, ptspg)) +
  geom_point(alpha = 0.5)# Prints the relationship between the new variables 'Defensive Rebounds Per Game' and 'Points Per Game' specific to players listed as Point Guards. 

reg_pg_drb_ppg

reg_pg_orb_ppg <- pg %>%
  mutate(orbpg = ORB / G,
         ptspg = PTS / G) %>%
  ggplot(aes(orbpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Offensive Rebounds Per Game' and 'Points Per Game' specific to players listed as Point Guards. 

reg_pg_orb_ppg

reg_pg_stl_ppg <- pg %>%
  mutate(stlpg = STL / G,
         ptspg = PTS / G) %>%
  ggplot(aes(stlpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Steals Per Game' and 'Points Per Game' specific to players listed as Point Guards. 

reg_pg_stl_ppg

reg_pg_ast_ppg <- pg %>%
  mutate(astpg = AST / G,
         ptspg = PTS / G) %>%
  ggplot(aes(astpg, ptspg)) +
  geom_point(alpha = 0.5)  # Prints the relationship between the new variables 'Assists Per Game' and 'Points Per Game' specific to players listed as Point Guards. 

reg_pg_ast_ppg # Prints the above formula

reg_pg_x3p_ppg <- pg %>%
  mutate(x3ppg = x3P / G,
         ptspg = PTS / G) %>%
  ggplot(aes(x3ppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables '3 Points Made Per Game' and 'Points Per Game' specific to players listed as Point Guards. 

reg_pg_x3p_ppg # Prints the above formula

reg_pg_x2p_ppg <- pg %>%
  mutate(x2ppg = x2P / G,
         ptspg = PTS / G) %>%
  ggplot(aes(x2ppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables '2 Points Made Per Game' and 'Points Per Game' specific to players listed as Point Guards. 

reg_pg_x2p_ppg # Prints the above formula

reg_pg_ft_ppg <- pg %>%
  mutate(ftppg = FT / G,
         ptspg = PTS / G) %>%
  ggplot(aes(ftppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Free Throws Made Per Game' and 'Points Per Game' specific to players listed as Point Guards. 

reg_pg_ft_ppg # Prints the above formula

Shooting Guard

reg_sg_drb_ppg <- sg %>%
  mutate(drbpg = DRB / G,
         ptspg = PTS / G) %>%
  ggplot(aes(drbpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Defensive Rebounds Per Game' and 'Points Per Game' specific to players listed as Shooting Guards. 

reg_sg_drb_ppg # Prints the above formula

reg_sg_orb_ppg <- sg %>%
  mutate(orbpg = ORB / G,
         ptspg = PTS / G) %>%
  ggplot(aes(orbpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Offensive Rebounds Per Game' and 'Points Per Game' specific to players listed as Shooting Guards.

reg_sg_orb_ppg # Prints the above formula

reg_sg_stl_ppg <- sg %>%
  mutate(stlpg = STL / G,
         ptspg = PTS / G) %>%
  ggplot(aes(stlpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Steals Per Game' and 'Points Per Game' specific to players listed as Shooting Guards.

reg_sg_stl_ppg # Prints the above formula

reg_sg_ast_ppg <- sg %>%
  mutate(astpg = AST / G,
         ptspg = PTS / G) %>%
  ggplot(aes(astpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Assists Per Game' and 'Points Per Game' specific to players listed as Shooting Guards.  

reg_sg_ast_ppg # Prints the above formula

reg_sg_x3p_ppg <- sg %>%
  mutate(x3ppg = x3P / G,
         ptspg = PTS / G) %>%
  ggplot(aes(x3ppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables '3 Points Made Per Game' and 'Points Per Game' specific to players listed as Shooting Guards.

reg_sg_x3p_ppg # Prints the above formula

reg_sg_x2p_ppg <- sg %>%
  mutate(x2ppg = x2P / G,
         ptspg = PTS / G) %>%
  ggplot(aes(x2ppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables '2 Points Made Per Game' and 'Points Per Game' specific to players listed as Shooting Guards.

reg_sg_x2p_ppg # Prints the above formula

reg_sg_ft_ppg <- sg %>%
  mutate(ftppg = FT / G,
         ptspg = PTS / G) %>%
  ggplot(aes(ftppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Free Throws Made Per Game' and 'Points Per Game' specific to players listed as Shooting Guards.

reg_sg_ft_ppg # Prints the above formula

Small Forward

reg_sf_drb_ppg <- sf %>%
  mutate(drbpg = DRB / G,
         ptspg = PTS / G) %>%
  ggplot(aes(drbpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Defensive Rebounds Per Game' and 'Points Per Game' specific to players listed as Small Forwards.

reg_sf_drb_ppg # Prints the above formula

reg_sf_orb_ppg <- sf %>%
  mutate(orbpg = ORB / G,
         ptspg = PTS / G) %>%
  ggplot(aes(orbpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Offensive Rebounds Per Game' and 'Points Per Game' specific to players listed as Small Forwards.

reg_sf_orb_ppg # Prints the above formula

reg_sf_stl_ppg <- sf %>%
  mutate(stlpg = STL / G,
         ptspg = PTS / G) %>%
  ggplot(aes(stlpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Steals Per Game' and 'Points Per Game' specific to players listed as Small Forwards.

reg_sf_stl_ppg # Prints the above formula

reg_sf_ast_ppg <- sf %>%
  mutate(astpg = AST / G,
         ptspg = PTS / G) %>%
  ggplot(aes(astpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Assists Per Game' and 'Points Per Game' specific to players listed as Small Forwards.  

reg_sf_ast_ppg # Prints the above formula

reg_sf_x3p_ppg <- sf %>%
  mutate(x3ppg = x3P / G,
         ptspg = PTS / G) %>%
  ggplot(aes(x3ppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables '3 Points Made Per Game' and 'Points Per Game' specific to players listed as Small Forwards.

reg_sf_x3p_ppg # Prints the above formula

reg_sf_x2p_ppg <- sf %>%
  mutate(x2ppg = x2P / G,
         ptspg = PTS / G) %>%
  ggplot(aes(x2ppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables '2 Points Made Per Game' and 'Points Per Game' specific to players listed as Small Forwards.

reg_sf_x2p_ppg # Prints the above formula

reg_sf_ft_ppg <- sf %>%
  mutate(ftppg = FT / G,
         ptspg = PTS / G) %>%
  ggplot(aes(ftppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Free Throws Made Per Game' and 'Points Per Game' specific to players listed as Small Forwards.

reg_sf_ft_ppg # Prints the above formula

Power Forward

reg_pf_drb_ppg <- pf %>%
  mutate(drbpg = DRB / G,
         ptspg = PTS / G) %>%
  ggplot(aes(drbpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Defensive Rebounds Per Game' and 'Points Per Game' specific to players listed as Power Forwards.

reg_pf_drb_ppg # Prints the above formula

reg_pf_orb_ppg <- pf %>%
  mutate(orbpg = ORB / G,
         ptspg = PTS / G) %>%
  ggplot(aes(orbpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Offensive Rebounds Per Game' and 'Points Per Game' specific to players listed as Power Forwards.

reg_pf_orb_ppg # Prints the above formula

reg_pf_stl_ppg <- pf %>%
  mutate(stlpg = STL / G,
         ptspg = PTS / G) %>%
  ggplot(aes(stlpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Steals Per Game' and 'Points Per Game' specific to players listed as Power Forwards.

reg_pf_stl_ppg # Prints the above formula

reg_pf_ast_ppg <- pf %>%
  mutate(astpg = AST / G,
         ptspg = PTS / G) %>%
  ggplot(aes(astpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Assists Per Game' and 'Points Per Game' specific to players listed as Power Forwards.  

reg_pf_ast_ppg # Prints the above formula

reg_pf_x3p_ppg <- pf %>%
  mutate(x3ppg = x3P / G,
         ptspg = PTS / G) %>%
  ggplot(aes(x3ppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables '3 Points Made Per Game' and 'Points Per Game' specific to players listed as Power Forwards.

reg_pf_x3p_ppg # Prints the above formula

reg_pf_x2p_ppg <- pf %>%
  mutate(x2ppg = x2P / G,
         ptspg = PTS / G) %>%
  ggplot(aes(x2ppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables '2 Points Made Per Game' and 'Points Per Game' specific to players listed as Power Forwards.

reg_pf_x2p_ppg # Prints the above formula

reg_pf_ft_ppg <- pf %>%
  mutate(ftppg = FT / G,
         ptspg = PTS / G) %>%
  ggplot(aes(ftppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Free Throws Made Per Game' and 'Points Per Game' specific to players listed as Power Forwards.

reg_pf_ft_ppg # Prints the above formula

Centres

reg_c_drb_ppg <- centres %>%
  mutate(drbpg = DRB / G,
         ptspg = PTS / G) %>%
  ggplot(aes(drbpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Defensive Rebounds Per Game' and 'Points Per Game' specific to players listed as Centres. 

reg_c_drb_ppg # Prints the above formula

reg_c_orb_ppg <- centres %>%
  mutate(orbpg = ORB / G,
         ptspg = PTS / G) %>%
  ggplot(aes(orbpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Offensive Rebounds Per Game' and 'Points Per Game' specific to players listed as Centres.

reg_c_orb_ppg # Prints the above formula

reg_c_stl_ppg <- centres %>%
  mutate(stlpg = STL / G,
         ptspg = PTS / G) %>%
  ggplot(aes(stlpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Steals Per Game' and 'Points Per Game' specific to players listed as Centres.

reg_c_stl_ppg # Prints the above formula

reg_c_ast_ppg <- centres %>%
  mutate(astpg = AST / G,
         ptspg = PTS / G) %>%
  ggplot(aes(astpg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Assists Per Game' and 'Points Per Game' specific to players listed as Centres.  

reg_c_ast_ppg # Prints the above formula

reg_c_x3p_ppg <- centres %>%
  mutate(x3ppg = x3P / G,
         ptspg = PTS / G) %>%
  ggplot(aes(x3ppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables '3 Points Made Per Game' and 'Points Per Game' specific to players listed as Centres.

reg_c_x3p_ppg # Prints the above formula

reg_c_x2p_ppg <- centres %>%
  mutate(x2ppg = x2P / G,
         ptspg = PTS / G) %>%
  ggplot(aes(x2ppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables '2 Points Made Per Game' and 'Points Per Game' specific to players listed as Centres.

reg_c_x2p_ppg # Prints the above formula

reg_c_ft_ppg <- centres %>%
  mutate(ftppg = FT / G,
         ptspg = PTS / G) %>%
  ggplot(aes(ftppg, ptspg)) +
  geom_point(alpha = 0.5) # Prints the relationship between the new variables 'Free Throws Made Per Game' and 'Points Per Game' specific to players listed as Centres.

reg_c_ft_ppg # Prints the above formula

Position specific five summary statistics

Point Guard

sum_stats_pg_drb_ppg <- pg %>%
  mutate(drbpg = DRB / G,
         ptspg = PTS / G) %>%
  summarize(avg_drb = mean(drbpg),
            s_drb = sd(drbpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(drbpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Defensive Rebounds Per Game and Points Per Game. It gives us the ability to predict the amount of points a Point Guard may make as well as the Defensive Rebounds made.

sum_stats_pg_orb_ppg <- pg %>%
  mutate(orbpg = ORB / G,
         ptspg = PTS / G) %>%
  summarize(avg_orb = mean(orbpg),
            s_orb = sd(orbpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(orbpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Offensive Rebounds Per Game and Points Per Game. It gives us the ability to predict the amount of points a Point Guard may make as well as the Offensive Rebounds made.

sum_stats_pg_stl_ppg <- pg %>%
  mutate(stlpg = STL / G,
         ptspg = PTS / G) %>%
  summarize(avg_stl = mean(stlpg),
            s_stl = sd(stlpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(stlpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Steals Per Game and Points Per Game. It gives us the ability to predict the amount of points a Point Guard may make as well as the Steals made.

sum_stats_pg_ast_ppg <- pg %>%
  mutate(astpg = AST / G,
         ptspg = PTS / G) %>%
  summarize(avg_ast = mean(astpg),
            s_ast = sd(astpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(astpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Assists Per Game and Points Per Game. It gives us the ability to predict the amount of points a Point Guard may make as well as the Assists made.

sum_stats_pg_x3p_ppg <- pg %>%
  mutate(x3ppg = x3P / G,
         ptspg = PTS / G) %>%
  summarize(avg_3p = mean(x3ppg),
            s_3p = sd(x3ppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(x3ppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team 3 Point Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Point Guard may make as well as the 3 point shots made.

sum_stats_pg_x2p_ppg <- pg %>%
  mutate(x2ppg = x2P / G,
         ptspg = PTS / G) %>%
  summarize(avg_2p = mean(x2ppg),
            s_2p = sd(x2ppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(x2ppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team 2 Point Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Point Guard may make as well as the 2 point shots made.

sum_stats_pg_ft_ppg <- pg %>%
  mutate(ftppg = FT / G,
         ptspg = PTS / G) %>%
  summarize(avg_ft = mean(ftppg),
            s_ft = sd(ftppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(ftppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Free Throw Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Point Guard may make as well as the Free Throw shots made.

Shooting Guard

sum_stats_sg_drb_ppg <- sg %>%
  mutate(drbpg = DRB / G,
         ptspg = PTS / G) %>%
  summarize(avg_drb = mean(drbpg),
            s_drb = sd(drbpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(drbpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Defensive Rebounds Per Game and Points Per Game. It gives us the ability to predict the amount of points a Shooting Guard may make as well as the Defensive Rebounds made.

sum_stats_sg_orb_ppg <- sg %>%
  mutate(orbpg = ORB / G,
         ptspg = PTS / G) %>%
  summarize(avg_orb = mean(orbpg),
            s_orb = sd(orbpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(orbpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Offensive Rebounds Per Game and Points Per Game. It gives us the ability to predict the amount of points a Shooting Guard may make as well as the Offensive Rebounds made.

sum_stats_sg_stl_ppg <- sg %>%
  mutate(stlpg = STL / G,
         ptspg = PTS / G) %>%
  summarize(avg_stl = mean(stlpg),
            s_stl = sd(stlpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(stlpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Steals Per Game and Points Per Game. It gives us the ability to predict the amount of points a Shooting Guard may make as well as the Steals made.

sum_stats_sg_ast_ppg <- sg %>%
  mutate(astpg = AST / G,
         ptspg = PTS / G) %>%
  summarize(avg_ast = mean(astpg),
            s_ast = sd(astpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(astpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Assists Per Game and Points Per Game. It gives us the ability to predict the amount of points a Shooting Guard may make as well as the Assists made.

sum_stats_sg_x3p_ppg <- sg %>%
  mutate(x3ppg = x3P / G,
         ptspg = PTS / G) %>%
  summarize(avg_3p = mean(x3ppg),
            s_3p = sd(x3ppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(x3ppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team 3 Point Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Shooting Guard may make as well as the 3 point shots made.

sum_stats_sg_x2p_ppg <- sg %>%
  mutate(x2ppg = x2P / G,
         ptspg = PTS / G) %>%
  summarize(avg_2p = mean(x2ppg),
            s_2p = sd(x2ppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(x2ppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team 2 Point Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Shooting Guard may make as well as the 2 point shots made.

sum_stats_sg_ft_ppg <- sg %>%
  mutate(ftppg = FT / G,
         ptspg = PTS / G) %>%
  summarize(avg_ft = mean(ftppg),
            s_ft = sd(ftppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(ftppg, ptspg))# Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Free Throw Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Shooting Guard may make as well as the Free Throw shots made.

Small Forward

sum_stats_sf_drb_ppg <- sf %>%
  mutate(drbpg = DRB / G,
         ptspg = PTS / G) %>%
  summarize(avg_drb = mean(drbpg),
            s_drb = sd(drbpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(drbpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Defensive Rebounds Per Game and Points Per Game. It gives us the ability to predict the amount of points a Small Forward may make as well as the Defensive Rebounds made.

sum_stats_sf_orb_ppg <- sf %>%
  mutate(orbpg = ORB / G,
         ptspg = PTS / G) %>%
  summarize(avg_orb = mean(orbpg),
            s_orb = sd(orbpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(orbpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Offensive Rebounds Per Game and Points Per Game. It gives us the ability to predict the amount of points a Small Forward may make as well as the Offensive Rebounds made.

sum_stats_sf_stl_ppg <- sf %>%
  mutate(stlpg = STL / G,
         ptspg = PTS / G) %>%
  summarize(avg_stl = mean(stlpg),
            s_stl = sd(stlpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(stlpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Steals Per Game and Points Per Game. It gives us the ability to predict the amount of points a Small Forward may make as well as the Steals made.

sum_stats_sf_ast_ppg <- sf %>%
  mutate(astpg = AST / G,
         ptspg = PTS / G) %>%
  summarize(avg_ast = mean(astpg),
            s_ast = sd(astpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(astpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Assists Per Game and Points Per Game. It gives us the ability to predict the amount of points a Small Forward may make as well as the Assists made.

sum_stats_sf_x3p_ppg <- sf %>%
  mutate(x3ppg = x3P / G,
         ptspg = PTS / G) %>%
  summarize(avg_3p = mean(x3ppg),
            s_3p = sd(x3ppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(x3ppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team 3 Point Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Small Forward may make as well as the 3 point shots made.

sum_stats_sf_x2p_ppg <- sf %>%
  mutate(x2ppg = x2P / G,
         ptspg = PTS / G) %>%
  summarize(avg_2p = mean(x2ppg),
            s_2p = sd(x2ppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(x2ppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team 2 Point Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Small Forward may make as well as the 2 point shots made.

sum_stats_sf_ft_ppg <- sf %>%
  mutate(ftppg = FT / G,
         ptspg = PTS / G) %>%
  summarize(avg_ft = mean(ftppg),
            s_ft = sd(ftppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(ftppg, ptspg))# Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Free Throw Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Small Forward may make as well as the Free Throw shots made.

Power Forward

sum_stats_pf_drb_ppg <- pf %>%
  mutate(drbpg = DRB / G,
         ptspg = PTS / G) %>%
  summarize(avg_drb = mean(drbpg),
            s_drb = sd(drbpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(drbpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Defensive Rebounds Per Game and Points Per Game. It gives us the ability to predict the amount of points a Power Forward may make as well as the Defensive Rebounds made.

sum_stats_pf_orb_ppg <- pf %>%
  mutate(orbpg = ORB / G,
         ptspg = PTS / G) %>%
  summarize(avg_orb = mean(orbpg),
            s_orb = sd(orbpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(orbpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Offensive Rebounds Per Game and Points Per Game. It gives us the ability to predict the amount of points a Power Forward may make as well as the Offensive Rebounds made.

sum_stats_pf_stl_ppg <- pf %>%
  mutate(stlpg = STL / G,
         ptspg = PTS / G) %>%
  summarize(avg_stl = mean(stlpg),
            s_stl = sd(stlpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(stlpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Steals Per Game and Points Per Game. It gives us the ability to predict the amount of points a Power Forward may make as well as the Steals made.

sum_stats_pf_ast_ppg <- pf %>%
  mutate(astpg = AST / G,
         ptspg = PTS / G) %>%
  summarize(avg_ast = mean(astpg),
            s_ast = sd(astpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(astpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Assists Per Game and Points Per Game. It gives us the ability to predict the amount of points a Power Forward may make as well as the Assists made.

sum_stats_pf_x3p_ppg <- pf %>%
  mutate(x3ppg = x3P / G,
         ptspg = PTS / G) %>%
  summarize(avg_3p = mean(x3ppg),
            s_3p = sd(x3ppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(x3ppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team 3 Point Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Power Forward may make as well as the 3 point shots made.

sum_stats_pf_x2p_ppg <- pf %>%
  mutate(x2ppg = x2P / G,
         ptspg = PTS / G) %>%
  summarize(avg_2p = mean(x2ppg),
            s_2p = sd(x2ppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(x2ppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team 2 Point Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Power Forward may make as well as the 2 point shots made.

sum_stats_pf_ft_ppg <- pf %>%
  mutate(ftppg = FT / G,
         ptspg = PTS / G) %>%
  summarize(avg_ft = mean(ftppg),
            s_ft = sd(ftppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(ftppg, ptspg))# Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Free Throw Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Power Forward may make as well as the Free Throw shots made.

Centres

sum_stats_c_drb_ppg <- centres %>%
  mutate(drbpg = DRB / G,
         ptspg = PTS / G) %>%
  summarize(avg_drb = mean(drbpg),
            s_drb = sd(drbpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(drbpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Defensive Rebounds Per Game and Points Per Game. It gives us the ability to predict the amount of points a Centres may make as well as the Defensive Rebounds made.

sum_stats_c_orb_ppg <- centres %>%
  mutate(orbpg = ORB / G,
         ptspg = PTS / G) %>%
  summarize(avg_orb = mean(orbpg),
            s_orb = sd(orbpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(orbpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Offensive Rebounds Per Game and Points Per Game. It gives us the ability to predict the amount of points a Centres may make as well as the Offensive Rebounds made.

sum_stats_c_stl_ppg <- centres %>%
  mutate(stlpg = STL / G,
         ptspg = PTS / G) %>%
  summarize(avg_stl = mean(stlpg),
            s_stl = sd(stlpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(stlpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Steals Per Game and Points Per Game. It gives us the ability to predict the amount of points a Centres may make as well as the Steals made.

sum_stats_c_ast_ppg <- centres %>%
  mutate(astpg = AST / G,
         ptspg = PTS / G) %>%
  summarize(avg_ast = mean(astpg),
            s_ast = sd(astpg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(astpg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Assists Per Game and Points Per Game. It gives us the ability to predict the amount of points a Centres may make as well as the Assists made.

sum_stats_c_x3p_ppg <- centres %>%
  mutate(x3ppg = x3P / G,
         ptspg = PTS / G) %>%
  summarize(avg_3p = mean(x3ppg),
            s_3p = sd(x3ppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(x3ppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team 3 Point Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Centres may make as well as the 3 point shots made.

sum_stats_c_x2p_ppg <- centres %>%
  mutate(x2ppg = x2P / G,
         ptspg = PTS / G) %>%
  summarize(avg_2p = mean(x2ppg),
            s_2p = sd(x2ppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(x2ppg, ptspg)) # Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team 2 Point Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Centres may make as well as the 2 point shots made.

sum_stats_c_ft_ppg <- centres %>%
  mutate(ftppg = FT / G,
         ptspg = PTS / G) %>%
  summarize(avg_ft = mean(ftppg),
            s_ft = sd(ftppg),
            avg_ptspg = mean(ptspg),
            s_ptspg = sd(ptspg),
            r = cor(ftppg, ptspg))# Creates a table visualising the summation of the five summary statistics. Mean and standard deviation of team Free Throw Makes Per Game and Points Per Game. It gives us the ability to predict the amount of points a Centres may make as well as the Free Throw shots made.

Regression Lines

Point Guards

regline_pg_drb_ppg <- sum_stats_pg_drb_ppg %>%
  summarise(slope = r * s_ptspg / s_drb,
            intercept = avg_ptspg - slope * avg_drb) # Use the previous five summary statistic and this formula to create the regression line

reg_pg_drb_ppg +
  geom_abline(intercept = regline_pg_drb_ppg$intercept, slope = regline_pg_drb_ppg$slope) # Plots the regression line on the graph

regline_pg_orb_ppg <- sum_stats_pg_orb_ppg %>%
  summarise(slope = r * s_ptspg / s_orb,
            intercept = avg_ptspg - slope * avg_orb) # Use the previous five summary statistic and this formula to create the regression line

reg_pg_orb_ppg +
  geom_abline(intercept = regline_pg_orb_ppg$intercept, slope = regline_pg_orb_ppg$slope) # Plots the regression line on the graph

regline_pg_stl_ppg <- sum_stats_pg_stl_ppg %>%
  summarise(slope = r * s_ptspg / s_stl,
            intercept = avg_ptspg - slope * avg_stl) # Use the previous five summary statistic and this formula to create the regression line

reg_pg_stl_ppg +
  geom_abline(intercept = regline_pg_stl_ppg$intercept, slope = regline_pg_stl_ppg$slope) # Plots the regression line on the graph

regline_pg_ast_ppg <- sum_stats_pg_ast_ppg %>%
  summarise(slope = r * s_ptspg / s_ast,
            intercept = avg_ptspg - slope * avg_ast) # Use the previous five summary statistic and this formula to create the regression line

reg_pg_ast_ppg +
  geom_abline(intercept = regline_pg_ast_ppg$intercept, slope = regline_pg_ast_ppg$slope) # Plots the regression line on the graph

regline_pg_x3p_ppg <- sum_stats_pg_x3p_ppg %>%
  summarise(slope = r * s_ptspg / s_3p,
            intercept = avg_ptspg - slope * avg_3p) # Use the previous five summary statistic and this formula to create the regression line

reg_pg_x3p_ppg +
  geom_abline(intercept = regline_pg_x3p_ppg$intercept, slope = regline_pg_x3p_ppg$slope, colour = "dodgerblue") # Plots the regression line on the graph

regline_pg_x2p_ppg <- sum_stats_pg_x2p_ppg %>%
  summarise(slope = r * s_ptspg / s_2p,
            intercept = avg_ptspg - slope * avg_2p) # Use the previous five summary statistic and this formula to create the regression line

reg_pg_x2p_ppg +
  geom_abline(intercept = regline_pg_x2p_ppg$intercept, slope = regline_pg_x2p_ppg$slope, colour = "orange") # Plots the regression line on the graph

regline_pg_ft_ppg <- sum_stats_pg_ft_ppg %>%
  summarise(slope = r * s_ptspg / s_ft,
            intercept = avg_ptspg - slope * avg_ft) # Use the previous five summary statistic and this formula to create the regression line

reg_pg_ft_ppg +
  geom_abline(intercept = regline_pg_ft_ppg$intercept, slope = regline_pg_ft_ppg$slope, colour = "pink") # Plots the regression line on the graph

Shooting Guard

regline_sg_drb_ppg <- sum_stats_sg_drb_ppg %>%
  summarise(slope = r * s_ptspg / s_drb,
            intercept = avg_ptspg - slope * avg_drb) # Use the previous five summary statistic and this formula to create the regression line

reg_sg_drb_ppg +
  geom_abline(intercept = regline_sg_drb_ppg$intercept, slope = regline_sg_drb_ppg$slope, colour = "green") # Plots the regression line on the graph

regline_sg_orb_ppg <- sum_stats_sg_orb_ppg %>%
  summarise(slope = r * s_ptspg / s_orb,
            intercept = avg_ptspg - slope * avg_orb) # Use the previous five summary statistic and this formula to create the regression line

reg_sg_orb_ppg +
  geom_abline(intercept = regline_sg_orb_ppg$intercept, slope = regline_sg_orb_ppg$slope) # Plots the regression line on the graph

regline_sg_stl_ppg <- sum_stats_sg_stl_ppg %>%
  summarise(slope = r * s_ptspg / s_stl,
            intercept = avg_ptspg - slope * avg_stl) # Use the previous five summary statistic and this formula to create the regression line

reg_sg_stl_ppg +
  geom_abline(intercept = regline_sg_stl_ppg$intercept, slope = regline_sg_stl_ppg$slope) # Plots the regression line on the graph

regline_sg_ast_ppg <- sum_stats_sg_ast_ppg %>%
  summarise(slope = r * s_ptspg / s_ast,
            intercept = avg_ptspg - slope * avg_ast) # Use the previous five summary statistic and this formula to create the regression line

reg_sg_ast_ppg +
  geom_abline(intercept = regline_sg_ast_ppg$intercept, slope = regline_sg_ast_ppg$slope) # Plots the regression line on the graph

regline_sg_x3p_ppg <- sum_stats_sg_x3p_ppg %>%
  summarise(slope = r * s_ptspg / s_3p,
            intercept = avg_ptspg - slope * avg_3p) # Use the previous five summary statistic and this formula to create the regression line

reg_sg_x3p_ppg +
  geom_abline(intercept = regline_sg_x3p_ppg$intercept, slope = regline_sg_x3p_ppg$slope) # Plots the regression line on the graph

regline_sg_x2p_ppg <- sum_stats_sg_x2p_ppg %>%
  summarise(slope = r * s_ptspg / s_2p,
            intercept = avg_ptspg - slope * avg_2p) # Use the previous five summary statistic and this formula to create the regression line

reg_sg_x2p_ppg +
  geom_abline(intercept = regline_sg_x2p_ppg$intercept, slope = regline_sg_x2p_ppg$slope) # Plots the regression line on the graph

regline_sg_ft_ppg <- sum_stats_sg_ft_ppg %>%
  summarise(slope = r * s_ptspg / s_ft,
            intercept = avg_ptspg - slope * avg_ft) # Use the previous five summary statistic and this formula to create the regression line

reg_sg_ft_ppg +
  geom_abline(intercept = regline_sg_ft_ppg$intercept, slope = regline_sg_ft_ppg$slope) # Plots the regression line on the graph

Small Forwards

regline_sf_drb_ppg <- sum_stats_sf_drb_ppg %>%
  summarise(slope = r * s_ptspg / s_drb,
            intercept = avg_ptspg - slope * avg_drb) # Use the previous five summary statistic and this formula to create the regression line

reg_sf_drb_ppg +
  geom_abline(intercept = regline_sf_drb_ppg$intercept, slope = regline_sf_drb_ppg$slope) # Plots the regression line on the graph

regline_sf_orb_ppg <- sum_stats_sf_orb_ppg %>%
  summarise(slope = r * s_ptspg / s_orb,
            intercept = avg_ptspg - slope * avg_orb) # Use the previous five summary statistic and this formula to create the regression line

reg_sf_orb_ppg +
  geom_abline(intercept = regline_sf_orb_ppg$intercept, slope = regline_sf_orb_ppg$slope) # Plots the regression line on the graph

regline_sf_stl_ppg <- sum_stats_sf_stl_ppg %>%
  summarise(slope = r * s_ptspg / s_stl,
            intercept = avg_ptspg - slope * avg_stl) # Use the previous five summary statistic and this formula to create the regression line

reg_sf_stl_ppg +
  geom_abline(intercept = regline_sf_stl_ppg$intercept, slope = regline_sf_stl_ppg$slope) # Plots the regression line on the graph

regline_sf_ast_ppg <- sum_stats_sf_ast_ppg %>%
  summarise(slope = r * s_ptspg / s_ast,
            intercept = avg_ptspg - slope * avg_ast) # Use the previous five summary statistic and this formula to create the regression line

reg_sf_ast_ppg +
  geom_abline(intercept = regline_sf_ast_ppg$intercept, slope = regline_sf_ast_ppg$slope) # Plots the regression line on the graph

regline_sf_x3p_ppg <- sum_stats_sf_x3p_ppg %>%
  summarise(slope = r * s_ptspg / s_3p,
            intercept = avg_ptspg - slope * avg_3p) # Use the previous five summary statistic and this formula to create the regression line

reg_sf_x3p_ppg +
  geom_abline(intercept = regline_sf_x3p_ppg$intercept, slope = regline_sf_x3p_ppg$slope) # Plots the regression line on the graph

regline_sf_x2p_ppg <- sum_stats_sf_x2p_ppg %>%
  summarise(slope = r * s_ptspg / s_2p,
            intercept = avg_ptspg - slope * avg_2p) # Use the previous five summary statistic and this formula to create the regression line

reg_sf_x2p_ppg +
  geom_abline(intercept = regline_sf_x2p_ppg$intercept, slope = regline_sf_x2p_ppg$slope) # Plots the regression line on the graph

regline_sf_ft_ppg <- sum_stats_sf_ft_ppg %>%
  summarise(slope = r * s_ptspg / s_ft,
            intercept = avg_ptspg - slope * avg_ft) # Use the previous five summary statistic and this formula to create the regression line

reg_sf_ft_ppg +
  geom_abline(intercept = regline_sf_ft_ppg$intercept, slope = regline_sf_ft_ppg$slope) # Plots the regression line on the graph

Centres

regline_c_drb_ppg <- sum_stats_c_drb_ppg %>%
  summarise(slope = r * s_ptspg / s_drb,
            intercept = avg_ptspg - slope * avg_drb) # Use the previous five summary statistic and this formula to create the regression line

reg_c_drb_ppg +
  geom_abline(intercept = regline_c_drb_ppg$intercept, slope = regline_c_drb_ppg$slope) # Plots the regression line on the graph

regline_c_orb_ppg <- sum_stats_c_orb_ppg %>%
  summarise(slope = r * s_ptspg / s_orb,
            intercept = avg_ptspg - slope * avg_orb) # Use the previous five summary statistic and this formula to create the regression line

reg_c_orb_ppg +
  geom_abline(intercept = regline_c_orb_ppg$intercept, slope = regline_c_orb_ppg$slope) # Plots the regression line on the graph

regline_c_stl_ppg <- sum_stats_c_stl_ppg %>%
  summarise(slope = r * s_ptspg / s_stl,
            intercept = avg_ptspg - slope * avg_stl) # Use the previous five summary statistic and this formula to create the regression line

reg_c_stl_ppg +
  geom_abline(intercept = regline_c_stl_ppg$intercept, slope = regline_c_stl_ppg$slope) # Plots the regression line on the graph

regline_c_ast_ppg <- sum_stats_c_ast_ppg %>%
  summarise(slope = r * s_ptspg / s_ast,
            intercept = avg_ptspg - slope * avg_ast) # Use the previous five summary statistic and this formula to create the regression line

reg_c_ast_ppg +
  geom_abline(intercept = regline_c_ast_ppg$intercept, slope = regline_c_ast_ppg$slope) # Plots the regression line on the graph

regline_c_x3p_ppg <- sum_stats_c_x3p_ppg %>%
  summarise(slope = r * s_ptspg / s_3p,
            intercept = avg_ptspg - slope * avg_3p) # Use the previous five summary statistic and this formula to create the regression line

reg_c_x3p_ppg +
  geom_abline(intercept = regline_c_x3p_ppg$intercept, slope = regline_c_x3p_ppg$slope) # Plots the regression line on the graph

regline_c_x2p_ppg <- sum_stats_c_x2p_ppg %>%
  summarise(slope = r * s_ptspg / s_2p,
            intercept = avg_ptspg - slope * avg_2p) # Use the previous five summary statistic and this formula to create the regression line

reg_c_x2p_ppg +
  geom_abline(intercept = regline_c_x2p_ppg$intercept, slope = regline_c_x2p_ppg$slope) # Plots the regression line on the graph

regline_c_ft_ppg <- sum_stats_c_ft_ppg %>%
  summarise(slope = r * s_ptspg / s_ft,
            intercept = avg_ptspg - slope * avg_ft) # Use the previous five summary statistic and this formula to create the regression line

reg_c_ft_ppg +
  geom_abline(intercept = regline_c_ft_ppg$intercept, slope = regline_c_ft_ppg$slope) # Plots the regression line on the graph

Linear Relationship for Points

Points = 3-Point FG + 2-Point FG + Free Throw

x3pts_ppg <- ggplot(data = p_stats, aes(x = x3P, y = PTS)) +
  geom_point(colour = "dodgerblue") +
  geom_smooth(method = "lm", colour = "magenta") + # 3pt FG makes to total points
  theme_linedraw()
  
x3pts_ppg # Prints the graph with a regression line

## `geom_smooth()` using formula 'y ~ x'

x2pts_ppg <- ggplot(data = p_stats, aes(x = x2P, y = PTS)) +
  geom_point(colour = "dodgerblue") +
  geom_smooth(method = "lm", colour = "magenta") + # 2pt FG makes to total points
  theme_light()

x2pts_ppg # Prints the graph with a regression line

## `geom_smooth()` using formula 'y ~ x'

ftpts_ppg <- ggplot(data = p_stats, aes(x = FT, y = PTS)) +
  geom_point(colour = "dodgerblue") +
  geom_smooth(method = "lm", colour = "magenta") +  # FT makes to total points
  theme_dark()
                
ftpts_ppg # Prints the graph with a regression line

## `geom_smooth()` using formula 'y ~ x'

Linear Modelling For PLayers

Combined Players

overall_tidy_combined <- lm(PTSpm ~ MP + x3P + x2P + FT, data = p_stats)

tidy(overall_tidy_combined, conf.int = TRUE) # Values players based on successful actions and their relative contribution to scoring points per minute.

## # A tibble: 5 x 7
##   term         estimate  std.error statistic   p.value  conf.low conf.high
##   <chr>           <dbl>      <dbl>     <dbl>     <dbl>     <dbl>     <dbl>
## 1 (Intercept)  0.382    0.00622        61.4  2.80e-198  0.370     0.394   
## 2 MP          -0.000187 0.00000924    -20.2  4.18e- 62 -0.000205 -0.000169
## 3 x3P          0.00140  0.0000778      18.0  9.47e- 53  0.00125   0.00155 
## 4 x2P          0.00105  0.0000538      19.6  1.97e- 59  0.000948  0.00116 
## 5 FT           0.000319 0.0000594       5.37 1.42e-  7  0.000202  0.000435

overall_tidy_ppg <- lm(PPG ~ G + x3P + x2P + FT, data = p_stats)

tidy(overall_tidy_ppg, conf.int = TRUE) # Values players based on successful actions and their relative contribution to scoring points.

## # A tibble: 5 x 7
##   term        estimate std.error statistic  p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept)   6.63     0.253       26.2  5.24e-87   6.13      7.12  
## 2 G            -0.100    0.00542    -18.5  8.34e-55  -0.111    -0.0896
## 3 x3P           0.0427   0.00150     28.4  8.04e-96   0.0397    0.0456
## 4 x2P           0.0297   0.00106     27.9  7.56e-94   0.0276    0.0317
## 5 FT            0.0136   0.00146      9.31 1.09e-18   0.0107    0.0165

overall_tidy_pts <- lm(PTS ~ AST + DRB + ORB + BLK + TOV, data = p_stats)

tidy(overall_tidy_pts, conf.int = TRUE)

## # A tibble: 6 x 7
##   term        estimate std.error statistic  p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept)   40.0      18.5        2.16 3.12e- 2    3.63    76.3   
## 2 AST           -0.272     0.181     -1.50 1.34e- 1   -0.627    0.0842
## 3 DRB            0.842     0.166      5.09 5.80e- 7    0.517    1.17  
## 4 ORB           -0.632     0.315     -2.00 4.58e- 2   -1.25    -0.0119
## 5 BLK            0.533     0.524      1.02 3.10e- 1   -0.498    1.56  
## 6 TOV            6.33      0.463     13.7  8.22e-35    5.42     7.24

Point Guard

pg_overall_tidy <- lm(PTSpm ~ MP + x3P + x2P + FT, data = pg)

tidy(pg_overall_tidy, conf.int = TRUE) # Values players based on successful actions and their relative contribution to scoring points per minute, specific to Point Guards. 

## # A tibble: 5 x 7
##   term         estimate std.error statistic  p.value   conf.low conf.high
##   <chr>           <dbl>     <dbl>     <dbl>    <dbl>      <dbl>     <dbl>
## 1 (Intercept)  0.363    0.0142        25.4  1.23e-38  0.334      0.391   
## 2 MP          -0.000162 0.0000196     -8.29 3.41e-12 -0.000201  -0.000123
## 3 x3P          0.00125  0.000150       8.33 2.83e-12  0.000952   0.00155 
## 4 x2P          0.00102  0.000112       9.13 8.48e-14  0.000800   0.00125 
## 5 FT           0.000264 0.000107       2.47 1.56e- 2  0.0000514  0.000477

pg_overall_tidy_ppg <- lm(PPG ~ G + x3P + x2P + FT, data = pg)

tidy(pg_overall_tidy_ppg, conf.int = TRUE) # Values players based on successful actions and their relative contribution to scoring points, specific to Point Guards

## # A tibble: 5 x 7
##   term        estimate std.error statistic  p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept)  7.10      0.630       11.3  8.59e-18  5.85       8.36  
## 2 G           -0.106     0.0136      -7.80 2.90e-11 -0.133     -0.0790
## 3 x3P          0.0423    0.00397     10.6  1.20e-16  0.0344     0.0502
## 4 x2P          0.0318    0.00252     12.7  2.70e-20  0.0268     0.0368
## 5 FT           0.00954   0.00306      3.11 2.61e- 3  0.00344    0.0156

pg_overall_tidy_pts <- lm(PTS ~ AST + DRB + ORB + BLK + TOV, data = pg)

tidy(pg_overall_tidy_pts, conf.int = TRUE) # Values players based on successful actions and their relative contribution to external actions not involved with points, specific to Point Guards.

## # A tibble: 6 x 7
##   term        estimate std.error statistic      p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>        <dbl>    <dbl>     <dbl>
## 1 (Intercept)   17.0      50.1       0.339 0.735         -82.8      117.  
## 2 AST            0.228     0.430     0.530 0.598          -0.628      1.08
## 3 DRB            0.152     0.616     0.246 0.806          -1.08       1.38
## 4 ORB           -1.15      2.11     -0.543 0.589          -5.36       3.06
## 5 BLK            2.85      3.12      0.911 0.365          -3.38       9.07
## 6 TOV            5.78      0.917     6.30  0.0000000191    3.95       7.61

Shooting Guard

sg_overall_tidy <- lm(PTSpm ~ MP + x3P + x2P + FT, data = sg)

tidy(sg_overall_tidy, conf.int = TRUE) # Values players based on successful actions and their relative contribution to scoring points per minute, specific to Shooting Guards.

## # A tibble: 5 x 7
##   term         estimate std.error statistic  p.value  conf.low conf.high
##   <chr>           <dbl>     <dbl>     <dbl>    <dbl>     <dbl>     <dbl>
## 1 (Intercept)  0.376    0.0119        31.6  1.77e-49  0.353     0.400   
## 2 MP          -0.000173 0.0000181     -9.56 3.19e-15 -0.000210 -0.000137
## 3 x3P          0.00145  0.000160       9.04 3.76e-14  0.00113   0.00177 
## 4 x2P          0.000842 0.000112       7.51 4.78e-11  0.000619  0.00107 
## 5 FT           0.000494 0.000154       3.21 1.87e- 3  0.000188  0.000800

sg_overall_tidy_ppg <- lm(PPG ~ G + x3P + x2P + FT, data = sg)

tidy(sg_overall_tidy_ppg, conf.int = TRUE) # Values players based on successful actions and their relative contribution to scoring points, specific to Shooting Guards.

## # A tibble: 5 x 7
##   term        estimate std.error statistic  p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept)   5.87     0.486       12.1  2.84e-20   4.90      6.83  
## 2 G            -0.0860   0.0109      -7.86 9.58e-12  -0.108    -0.0643
## 3 x3P           0.0428   0.00312     13.7  1.96e-23   0.0365    0.0490
## 4 x2P           0.0253   0.00258      9.82 9.48e-16   0.0202    0.0304
## 5 FT            0.0190   0.00382      4.98 3.13e- 6   0.0115    0.0266

sg_overall_tidy_pts <- lm(PTS ~ AST + DRB + ORB + BLK + TOV, data = sg)

tidy(sg_overall_tidy_pts, conf.int = TRUE) # Values players based on successful actions and their relative contribution to external actions not involved with points, specific to Shooting Guards.

## # A tibble: 6 x 7
##   term        estimate std.error statistic    p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>      <dbl>    <dbl>     <dbl>
## 1 (Intercept)  -13.8      38.5      -0.359 0.720       -90.4       62.7 
## 2 AST            0.246     0.474     0.519 0.605        -0.696      1.19
## 3 DRB            2.20      0.462     4.76  0.00000781    1.28       3.11
## 4 ORB            1.99      1.46      1.36  0.177        -0.912      4.88
## 5 BLK           -3.29      2.20     -1.49  0.139        -7.67       1.09
## 6 TOV            4.32      0.946     4.56  0.0000167     2.44       6.20

Small Forward

sf_overall_tidy <- lm(PTSpm ~ MP + x3P + x2P + FT, data = sf)

tidy(sf_overall_tidy, conf.int = TRUE) # Values players based on successful actions and their relative contribution to scoring points per minute, specific to Small Forwards.

## # A tibble: 5 x 7
##   term         estimate std.error statistic  p.value   conf.low conf.high
##   <chr>           <dbl>     <dbl>     <dbl>    <dbl>      <dbl>     <dbl>
## 1 (Intercept)  0.346    0.0119        29.0  1.25e-34  0.322      0.370   
## 2 MP          -0.000163 0.0000192     -8.49 1.63e-11 -0.000201  -0.000124
## 3 x3P          0.00131  0.000230       5.69 5.36e- 7  0.000848   0.00177 
## 4 x2P          0.00106  0.000132       8.03 8.68e-11  0.000794   0.00132 
## 5 FT           0.000294 0.000152       1.94 5.82e- 2 -0.0000106  0.000600

sf_overall_tidy_ppg <- lm(PPG ~ G + x3P + x2P + FT, data = sf)

tidy(sf_overall_tidy_ppg, conf.int = TRUE) # Values players based on successful actions and their relative contribution to scoring points, specific to Small Forwards

## # A tibble: 5 x 7
##   term        estimate std.error statistic  p.value  conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>     <dbl>     <dbl>
## 1 (Intercept)  7.34      0.569       12.9  4.05e-18  6.20        8.48  
## 2 G           -0.117     0.0125      -9.36 6.69e-13 -0.143      -0.0923
## 3 x3P          0.0438    0.00535      8.18 5.03e-11  0.0331      0.0545
## 4 x2P          0.0364    0.00335     10.9  3.37e-15  0.0296      0.0431
## 5 FT           0.00819   0.00456      1.80 7.79e- 2 -0.000946    0.0173

sf_overall_tidy_pts <- lm(PTS ~ AST + DRB + ORB + BLK + TOV, data = sf)

tidy(sf_overall_tidy_pts, conf.int = TRUE) # Values players based on successful actions and their relative contribution to external actions not involved with points, specific to Small Forwards

## # A tibble: 6 x 7
##   term        estimate std.error statistic   p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>     <dbl>    <dbl>     <dbl>
## 1 (Intercept)  -73.4      43.6      -1.68  0.0982    -161.       14.1  
## 2 AST           -0.642     0.532    -1.21  0.233       -1.71      0.425
## 3 DRB            1.77      0.504     3.51  0.000920     0.758     2.78 
## 4 ORB           -0.497     1.06     -0.468 0.642       -2.63      1.63 
## 5 BLK            0.833     1.58      0.528 0.600       -2.33      4.00 
## 6 TOV            6.12      1.31      4.67  0.0000212    3.49      8.75

Power Forward

pf_overall_tidy <- lm(PTSpm ~ MP + x3P + x2P + FT, data = pf)

tidy(pf_overall_tidy, conf.int = TRUE) # Values players based on successful actions and their relative contribution to scoring points per minute, specific to Power Forwards.

## # A tibble: 5 x 7
##   term         estimate std.error statistic  p.value  conf.low conf.high
##   <chr>           <dbl>     <dbl>     <dbl>    <dbl>     <dbl>     <dbl>
## 1 (Intercept)  0.413    0.0177        23.4  3.03e-34  0.378     0.448   
## 2 MP          -0.000246 0.0000345     -7.13 8.41e-10 -0.000314 -0.000177
## 3 x3P          0.00184  0.000395       4.66 1.54e- 5  0.00105   0.00263 
## 4 x2P          0.00125  0.000207       6.06 6.79e- 8  0.000840  0.00167 
## 5 FT           0.000295 0.000238       1.24 2.19e- 1 -0.000179  0.000770

pf_overall_tidy_ppg <- lm(PPG ~ G + x3P + x2P + FT, data = pf)

tidy(pf_overall_tidy_ppg, conf.int = TRUE) # Values players based on successful actions and their relative contribution to scoring points, specific to Power Forwards. 

## # A tibble: 5 x 7
##   term        estimate std.error statistic  p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept)   6.24     0.641        9.74 1.61e-14  4.96       7.52  
## 2 G            -0.101    0.0155      -6.51 1.07e- 8 -0.132     -0.0700
## 3 x3P           0.0479   0.00564      8.49 2.79e-12  0.0366     0.0592
## 4 x2P           0.0288   0.00314      9.16 1.74e-13  0.0225     0.0351
## 5 FT            0.0152   0.00454      3.34 1.35e- 3  0.00612    0.0242

pf_overall_tidy_pts <- lm(PTS ~ AST + DRB + ORB + BLK + TOV, data = pf)

tidy(pf_overall_tidy_pts, conf.int = TRUE) # Values players based on successful actions and their relative contribution to external actions not involved with points, specific to Power Fowards

## # A tibble: 6 x 7
##   term        estimate std.error statistic  p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept) -23.2       30.1     -0.772  4.43e- 1  -83.3      36.8  
## 2 AST          -1.51       0.395   -3.83   2.81e- 4   -2.30     -0.725
## 3 DRB           1.25       0.257    4.88   6.96e- 6    0.741     1.77 
## 4 ORB           0.0810     0.544    0.149  8.82e- 1   -1.01      1.17 
## 5 BLK           0.0720     0.929    0.0775 9.38e- 1   -1.78      1.93 
## 6 TOV           6.74       0.910    7.40   2.84e-10    4.92      8.55

Centres

c_overall_tidy <- lm(PTSpm ~ MP + x3P + x2P + FT, data = centres)

tidy(c_overall_tidy, conf.int = TRUE) # Values players based on successful actions and their relative contribution to scoring points per minute, specific to Centres.

## # A tibble: 5 x 7
##   term         estimate std.error statistic  p.value  conf.low conf.high
##   <chr>           <dbl>     <dbl>     <dbl>    <dbl>     <dbl>     <dbl>
## 1 (Intercept)  0.400    0.0122        32.8  3.56e-42  0.376     0.425   
## 2 MP          -0.000225 0.0000210    -10.7  4.89e-16 -0.000267 -0.000183
## 3 x3P          0.00175  0.000198       8.82 1.03e-12  0.00135   0.00214 
## 4 x2P          0.00116  0.000107      10.8  3.19e-16  0.000947  0.00137 
## 5 FT           0.000367 0.000108       3.38 1.22e- 3  0.000150  0.000584

c_overall_tidy_ppg <- lm(PPG ~ G + x3P + x2P + FT, data = centres)

tidy(c_overall_tidy_ppg, conf.int = TRUE) # Values players based on successful actions and their relative contribution to scoring points, specific to Centres.

## # A tibble: 5 x 7
##   term        estimate std.error statistic  p.value conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
## 1 (Intercept)   7.35     0.568       12.9  1.17e-19   6.22      8.49  
## 2 G            -0.115    0.0116      -9.90 1.31e-14  -0.138    -0.0919
## 3 x3P           0.0433   0.00452      9.58 4.77e-14   0.0343    0.0524
## 4 x2P           0.0282   0.00192     14.7  2.13e-22   0.0244    0.0321
## 5 FT            0.0177   0.00286      6.20 4.36e- 8   0.0120    0.0234

c_overall_tidy_pts <- lm(PTS ~ AST + DRB + ORB + BLK + TOV, data = centres)

tidy(c_overall_tidy_pts, conf.int = TRUE) # Values players based on successful actions and their relative contribution to external actions not involved with points, specific to Centres

## # A tibble: 6 x 7
##   term        estimate std.error statistic    p.value  conf.low conf.high
##   <chr>          <dbl>     <dbl>     <dbl>      <dbl>     <dbl>     <dbl>
## 1 (Intercept) -58.3       36.8     -1.58   0.118      -132.        15.3  
## 2 AST           0.662      0.455    1.46   0.150        -0.246      1.57 
## 3 DRB           0.0150     0.269    0.0556 0.956        -0.523      0.552
## 4 ORB           0.654      0.367    1.78   0.0798       -0.0798     1.39 
## 5 BLK           2.44       0.534    4.57   0.0000231     1.37       3.50 
## 6 TOV           5.06       0.982    5.16   0.00000264    3.10       7.03

#Player Points Predictions by Position

Point Guard

pg_ppg_pts_predict <- pg %>%
  filter(PPG >= 10) %>% # This will filter out Point Guards who score under 10 Points Per Game
  mutate(pg_r_hat = predict(pg_overall_tidy_ppg, newdata = .)) %>%
  ggplot(aes(pg_r_hat, PTS, label = player_name)) +
  geom_point() +
  geom_text(nudge_x = 2, cex = 2) +
  geom_smooth() # This formula will predict each Point Guards predicted Points Per Game against their actual output 

pg_ppg_pts_predict # Prints the output

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Shooting Guard

sg_ppg_pts_predict <- sg %>%
  filter(PPG >= 10) %>% # This will filter out Shooting Guards who score under 10 Points Per Game
  mutate(sg_r_hat = predict(sg_overall_tidy_ppg, newdata = .)) %>%
  ggplot(aes(sg_r_hat, PTS, label = player_name)) +
  geom_point() +
  geom_text(nudge_x = 2, cex = 2) +
  geom_smooth() # This formula will predict each Shooting Guards predicted Points Per Game against their actual output

sg_ppg_pts_predict # Prints the output

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Small Forward

sf_ppg_pts_predict <- sf %>%
  filter(PPG >= 10) %>% # This will filter out Small Forwards who score under 10 Points Per Game
  mutate(sf_r_hat = predict(sf_overall_tidy_ppg, newdata = .)) %>%
  ggplot(aes(sf_r_hat, PTS, label = player_name)) +
  geom_point() +
  geom_text(nudge_x = 2, cex = 2) +
  geom_smooth() # This formula will predict each Small Forwards predicted Points Per Game against their actual output

sf_ppg_pts_predict # Prints the output

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Power Forward

pf_ppg_pts_predict <- pf %>%
  filter(PPG >= 10) %>% # This will filter out Power Forwards who score under 10 Points Per Game
  mutate(pf_r_hat = predict(pf_overall_tidy_ppg, newdata = .)) %>%
  ggplot(aes(pf_r_hat, PTS, label = player_name)) +
  geom_point(colour = "Red", alpha = 0.8) +
  geom_text(nudge_x = 2, cex = 2) +
  geom_smooth() + # This formula will predict each Power Forwards predicted Points Per Game against their actual output
  labs(title = "Successful Free Throw Point Attempts and Overall Points",
       subtitle = "Identifying a Relationship between Free Throws and Total Points",
       x = "Successful Free Throws",
       y = "Total Points") + # Changing and configuring the Titles of the graph. 
  theme_linedraw()
  
pf_ppg_pts_predict # Prints the output

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Centres

c_ppg_pts_predict <- centres %>%
  filter(PPG >= 10) %>% # This will filter out Centres who score under 10 Points Per Game
  mutate(c_r_hat = predict(c_overall_tidy_ppg, newdata = .)) %>%
  ggplot(aes(c_r_hat, PTS, label = player_name)) +
  geom_point(colour = "Red", alpha = 0.8) +
  geom_text(nudge_x = 2, cex = 2) +
  geom_smooth() # This formula will predict each Centres predicted Points Per Game against their actual output 

c_ppg_pts_predict # Prints the output

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Adding Salary values

Point Guard

pg_pts_hat <- pg %>%
  mutate(pg_hat = predict(pg_overall_tidy_ppg, newdata = .))

pg_salary <- pg_pts_hat %>%
  ggplot(aes(Salary, pg_hat, colour = Pos, label = player_name)) +
  geom_point() +
  geom_text(nudge_y = 1, cex = 2) # Compares and plots individual Player Salaries against their predicted Points Per Game to identify value
  
  
pg_salary # Gives the predicted points average against the players salary, to see where the value is. We select D'Angelo Russell or Kemba Walker at Point Guard. 

Shooting Guard

sg_pts_hat <- sg %>%
  mutate(sg_hat = predict(sg_overall_tidy_ppg, newdata = .))

sg_salary <- sg_pts_hat %>%
  ggplot(aes(Salary, sg_hat, colour = Pos, label = player_name)) +
  geom_point() +
  geom_text(nudge_y = 1.5, cex = 2) # Compares and plots individual Player Salaries against their predicted Points Per Game to identify value
  
  
sg_salary # Gives the predicted points average against the players salary, to see where the value is. We select Donovan Mitchell or Devin Booker at Shooting Guard. 

Small Forward

sf_pts_hat <- sf %>%
  mutate(sf_hat = predict(sf_overall_tidy_ppg, newdata = .))

sf_salary <- sf_pts_hat %>%
  ggplot(aes(Salary, sf_hat, colour = Pos, label = player_name)) +
  geom_point() +
  geom_text(nudge_y = 1.5, cex = 2) # Compares and plots individual Player Salaries against their predicted Points Per Game to identify value
  
  
sf_salary # Gives the predicted points average against the players salary, to see where the value is. Depending on salary space, we either pick Kevin Durant or Kawhi Leonard at Small Forward as they stand out from the pack of lower priced players.

Power Forward

pf_pts_hat <- pf %>%
  mutate(pf_hat = predict(pf_overall_tidy_ppg, newdata = .))

pf_salary <- pf_pts_hat %>%
  ggplot(aes(Salary, pf_hat, colour = Pos, label = player_name)) +
  geom_point() +
  geom_text(nudge_y = 1.5, cex = 2) # Compares and plots individual Player Salaries against their predicted Points Per Game to identify value
  
  
pf_salary # Gives the predicted points average against the players salary, to see where the value is. We select either Julius Randle or Tobias Harris at the Power Forward position. 

Centres

c_pts_hat <- centres %>%
  mutate(c_hat = predict(c_overall_tidy_ppg, newdata = .))

c_salary <- c_pts_hat %>%
  ggplot(aes(Salary, c_hat, colour = Pos, label = player_name)) +
  geom_point() +
  geom_text(nudge_y = 1.5, cex = 2) # Compares and plots individual Player Salaries against their predicted Points Per Game to identify value
  
  
c_salary # Gives the predicted points average against the players salary, to see where the value is. We select Karl-Anthony Towns at Centre. 

Selected Team

selected_player_pg <- pg %>%
  filter(player_name == "D'Angelo Russell") # Filter for the chosen PG

selected_player_pg_2 <- pg %>%
  filter(player_name == "Kemba Walker") # Filter for the chosen PG

selected_pg <- bind_rows(selected_player_pg, selected_player_pg_2) # Combine the two selected PG players for a comparison to make a final decision. 

selected_player_sg <- sg %>%
  filter(player_name == "Donovan Mitchell") # Filter for the chosen SG

selected_player_sg_2 <- sg %>%
  filter(player_name == "Devin Booker") # Filter for the chosen SG

selected_sg <- bind_rows(selected_player_sg, selected_player_sg_2) # Combine the two selected SG players for a comparison to make a final decision. 

selected_player_sf <- sf %>%
  filter(player_name == "Kevin Durant") # Filter for the chosen SF

selected_player_sf_2 <- sf %>%
  filter(player_name == "Kawhi Leonard") # Filter for the chosen SF

selected_sf <- bind_rows(selected_player_sf, selected_player_sf_2) # Combine the two selected SF players for a comparison to make a final decision. 

selected_player_pf <- pf %>%
  filter(player_name == "Giannis Antetokounmpo") # Filter for the chosen PF

selected_player_pf_2 <- pf %>%
  filter(player_name == "Julius Randle") # Filter for the chosen PF

selected_pf <- bind_rows(selected_player_pf, selected_player_pf_2) # Combine the two selected PF players for a comparison to make a final decision. 

selected_player_c <- centres %>%
  filter(player_name == "Karl-Anthony Towns") # Filter for the chosen C

Final Graphs For Communication/Report

# Offence Rating v Wins

final_of_wins <- comb_team %>%
  ggplot(aes(ORtg, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "blue") +
  labs(title = "Offensive Rating relative to Wins",
       subtitle = "A Linear Relationship Explored",
       caption = "Plotting Team Offensive Ratings and their correlating Wins",
       x = "Team Offensive Ratings",
       y = "Season Wins") # Changing and configuring the Titles of the graph. 
  
of_wins

## `geom_smooth()` using formula 'y ~ x'

# Defence Rating v Wins
  
final_def_wins <- comb_team %>%
  ggplot(aes(DRtg, W)) +
  geom_point() +
  geom_smooth(method = "lm", colour = "orange") +
  labs(title = "Defensive Rating relative to Wins",
       subtitle = "A Linear Relationship",
       caption = "Plotting Team Defensive Ratings and their correlating Wins",
       x = "Team Defensive Ratings",
       y = "Season Wins") # Changing and configuring the Titles of the graph. 
   # Testing for a linear relationship between Defensive Rating and Wins. Important to note that the negative linear relationship is actually reversed, as a higher defensive rating is not a good outcome for the team. A higher win rate and lower defensive rating is the ideal outcome.

def_wins  

## `geom_smooth()` using formula 'y ~ x'

# The Offence Residuals
  
final_of_points <- ggplot(data = NULL, aes(x = of_wins_points, y = of_wins_stdres)) +
  geom_point(colour = "red") +
  geom_text(aes(label = of_wins_labels), nudge_y = 0.3) +
  ylim(c(-4, 4)) +
  geom_hline(yintercept = c(-3, 3), colour = "darkgreen", linetype = "dashed") +
  labs(title = "Offensive Rating - Potential Leverage",
       subtitle = "A Linear Relationship Explored",
       caption = "Plotting Team Offensive Ratings and their leverage potential",
       x = "Points of Offensive Rating Residuals",
       y = "Standardised Residuals") # Changing and configuring the Titles of the graph. 

final_of_points

# Leverage of Offensive Rating

final_of_lev <- ggplot(data = NULL, aes(x = of_wins_points, y = of_wins_hats)) +
  geom_point(colour = "dodgerblue") +
  labs(title = "Measuring Offensive Rating Leverage relative to Wins",
       subtitle = "A Linear Relationship Explored",
       caption = "Measuring hat values to determine leverage points",
       x = "Points of Offensive Rating Residuals",
       y = "Hat Values of Linear Model") # Changing and configuring the Titles of the graph. 

final_of_lev

# Collective Change in the Coefficients of Offensive Rating and Wins

final_of_cook <- ggplot(data = NULL, aes(x = of_wins_points, y = of_wins_cook))+
  geom_point(colour = "tomato2") +
  labs(title = "Collective Change in Coefficients of Offensive Rating and Wins",
       subtitle = "A Linear Relationship Explored",
       caption = "Plotting the Coefficient Changes",
       x = "Points of Offensive Rating Residuals",
       y = "Change in Coefficient") # Changing and configuring the Titles of the graph. 

final_of_cook

# Homoscedasticity

final_of_h <- ggplot(data = NULL, aes(x = of_wins_fitted, y = of_wins_res)) +
  geom_point(colour = "dodgerblue") +
  geom_hline(yintercept = 0, colour = "red", linetype = "dashed") +
  labs(title = "Offensive Rating relative to Wins",
       subtitle = "A Linear Relationship Explored",
       caption = "Measuring the constant variance of Predicted Wins",
       x = "Predicted Wins Relative to Offensive Rating",
       y = "Residuals of Wins Relative to Offensive Rating") # Changing and configuring the Titles of the graph. 

final_of_h

# Value in Predicted Output

## Centres

final_c_salary <- c_pts_hat %>%
  ggplot(aes(Salary, c_hat, colour = Pos, label = player_name)) +
  geom_point() +
  geom_text(nudge_y = 1.5, cex = 2) + # Gives the predicted points average against the players salary, to see where the value is. We select Karl-Anthony Towns at Centre. 
  geom_hline(yintercept = 20, colour = "black", linetype = "dotted") +
  geom_vline(xintercept = 15000000, colour = "black", linetype = "dotted") + # Divides the graph into sections to indicate where players may be overpriced for their output
  labs(title = "Predicted Points and Current Salary for Centres",
        subtitle = "Finding Value in the Centre Position",
        caption = "Finding a cheaper alternative with similar predicted output",
        x = "Player Salary",
        y = "Predicted Points Per Game",
       colour = "Player Position")

final_c_salary

## Power Forward

final_pf_salary <- pf_pts_hat %>%
  ggplot(aes(Salary, pf_hat, colour = Pos, label = player_name)) +
  geom_point() +
  geom_text(nudge_y = 1.5, cex = 2) + # Gives the predicted points average against the players salary, to see where the value is. We select either Julius Randle or Tobias Harris at the Power Forward position. 
  geom_hline(yintercept = 20, colour = "dodgerblue", linetype = "dotted") +
  geom_vline(xintercept = 15000000, colour = "dodgerblue", linetype = "dotted") + # Divides the graph into sections to indicate where players may be overpriced for their output
  labs(title = "Predicted Points and Current Salary for Power Forwards",
       subtitle = "Finding Value in the Power Forward Position",
       caption = "Finding a cheaper alternative with similar predicted output",
       x = "Player Salary",
       y = "Predicted Points Per Game",
       colour = "Player Position")

final_pf_salary

## Small Forward

final_sf_salary <- sf_pts_hat %>%
  ggplot(aes(Salary, sf_hat, colour = Pos, label = player_name)) +
  geom_point() +
  geom_text(nudge_y = 1.5, cex = 2) + # Gives the predicted points average against the players salary, to see where the value is. Depending on salary space, we either pick Kevin Durant or Kawhi Leonard at Small Forward as they stand out from the pack of lower priced players.
  geom_hline(yintercept = 20, colour = "darkorchid2", linetype = "dotted") +
  geom_vline(xintercept = 15000000, colour = "darkorchid2", linetype = "dotted") + # Divides the graph into sections to indicate where players may be overpriced for their output
  labs(title = "Predicted Points and Current Salary for Small Forwards",
       subtitle = "Finding Value in the Small Forward Position",
       caption = "Finding a cheaper alternative with similar predicted output",
       x = "Player Salary",
       y = "Predicted Points Per Game",
       colour = "Player Position")

final_sf_salary

## Shooting Guard

final_sg_salary <- sg_pts_hat %>%
  ggplot(aes(Salary, sg_hat, colour = Pos, label = player_name)) +
  geom_point() +
  geom_text(nudge_y = 1.5, cex = 2) + # Gives the predicted points average against the players salary, to see where the value is. We select Donovan Mitchell at Shooting Guard. 
  geom_hline(yintercept = 20, colour = "cyan", linetype = "dotted") +
  geom_vline(xintercept = 15000000, colour = "cyan", linetype = "dotted") + # Divides the graph into sections to indicate where players may be overpriced for their output
  labs(title = "Predicted Points and Current Salary for Shooting Guards",
       subtitle = "Finding Value in the Shooting Guard Position",
       caption = "Finding a cheaper alternative with similar predicted output",
       x = "Player Salary",
       y = "Predicted Points Per Game",
       colour = "Player Position")

final_sg_salary

## Point Guard

final_pg_salary <- pg_pts_hat %>%
  ggplot(aes(Salary, pg_hat, colour = Pos, label = player_name)) +
  geom_point() +
  geom_text(nudge_y = 1, cex = 2) + # Gives the predicted points average against the players salary, to see where the value is. We select D'Angelo Russell at Point Guard. 
  geom_hline(yintercept = 20, colour = "navy", linetype = "dotted") +
  geom_vline(xintercept = 15000000, colour = "navy", linetype = "dotted") + # Divides the graph into sections to indicate where players may be overpriced for their output
  labs(title = "Predicted Points and Current Salary for Point Guards",
       subtitle = "Finding Value in the Point Guard Position",
       caption = "Finding a cheaper alternative with similar predicted output",
       x = "Player Salary",
       y = "Predicted Points Per Game",
       colour = "Player Position")

final_pg_salary

## Linear Relationship for Points 

# Points = 3-Point FG + 2-Point FG + Free Throw

# 3 Point Relationship

final_x3pts_ppg <- ggplot(data = p_stats, aes(x = x3P, y = PTS)) +
  geom_point(colour = "dodgerblue") +
  geom_smooth(method = "lm", colour = "magenta") + # 3pt FG makes to total points
  labs(title = "Successful 3 Point Attempts and Overall Points",
       subtitle = "Identifying a Relationship between 3 Point Baskets and Total Points",
       x = "Successful 3 Point Baskets",
       y = "Total Points") + # Changing and configuring the Titles of the graph. 
  theme_linedraw()
  
  
# 2 Point Relationship

final_x2pts_ppg <- ggplot(data = p_stats, aes(x = x2P, y = PTS)) +
  geom_point(colour = "dodgerblue") +
  geom_smooth(method = "lm", colour = "magenta") + # 2pt FG makes to total points
  labs(title = "Successful 2 Point Attempts and Overall Points",
     subtitle = "Identifying a Relationship between 2 Point Baskets and Total Points",
     x = "Successful 2 Point Baskets",
     y = "Total Points") + # Changing and configuring the Titles of the graph. 
  theme_linedraw()


# Free Throw Relationship

final_ftpts_ppg <- ggplot(data = p_stats, aes(x = FT, y = PTS)) +
  geom_point(colour = "dodgerblue") +
  geom_smooth(method = "lm", colour = "magenta") + # FT makes to total points
  labs(title = "Successful Free Throw Point Attempts and Overall Points",
       subtitle = "Identifying a Relationship between Free Throws and Total Points",
       x = "Successful Free Throws",
       y = "Total Points") + # Changing and configuring the Titles of the graph. 
  theme_linedraw()

# Points Per Game Value

combined_ppgsalary_value <- p_stats %>%
  ggplot(aes(Salary, PPG, colour = Pos, label = player_name)) +
  geom_point() +
  geom_text(nudge_y = 1, cex = 2) +
  geom_hline(yintercept = 20, colour = "navy", linetype = "dotted") +
  geom_vline(xintercept = 15000000, colour = "navy", linetype = "dotted") +
  scale_y_continuous(limits = c(10,35)) +
    labs(title = "Current Points Per Game",
       subtitle = "Finding Value in the NBA",
       caption = "Finding a cheaper alternative with similar predicted output",
       x = "Player Salary",
       y = "Points Per Game",
       colour = "Player Position") # Changing and configuring the Titles of the graph. 

combined_ppgsalary_value

## Warning: Removed 234 rows containing missing values (geom_point).

## Warning: Removed 234 rows containing missing values (geom_text).

# Rebounds Per Game Value

combined_rpgsalary_value <- p_stats %>%
  ggplot(aes(Salary, RPG, colour = Pos, label = player_name)) +
  geom_point() +
  geom_text(nudge_y = .5, cex = 2) +
  geom_hline(yintercept = 7, colour = "navy", linetype = "dotted") +
  geom_vline(xintercept = 15000000, colour = "navy", linetype = "dotted") +
  scale_y_continuous(limits = c(5,17)) +
    labs(title = "Current Total Rebounds Per Game",
       subtitle = "Finding Value in the NBA",
       caption = "Finding a cheaper alternative with similar predicted output",
       x = "Player Salary",
       y = "Rebounds Per Game",
       colour = "Player Position") # Changing and configuring the Titles of the graph. 

combined_rpgsalary_value

## Warning: Removed 284 rows containing missing values (geom_point).

## Warning: Removed 284 rows containing missing values (geom_text).

# Assists Per Game Value

combined_astsalary_value <- p_stats %>%
  ggplot(aes(Salary, APG, colour = Pos, label = player_name)) +
  geom_point() +
  geom_text(nudge_y = .5, cex = 2) +
  geom_hline(yintercept = 7, colour = "navy", linetype = "dotted") +
  geom_vline(xintercept = 15000000, colour = "navy", linetype = "dotted") +
  scale_y_continuous(limits = c(4,12)) +
    labs(title = "Current Assists Per Game",
       subtitle = "Finding Value in the NBA",
       caption = "Finding a cheaper alternative with similar predicted output",
       x = "Player Salary",
       y = "Assists Per Game",
       colour = "Player Position") + # Changing and configuring the Titles of the graph.
  theme_bw() # Change the theme of the graph

combined_astsalary_value

## Warning: Removed 330 rows containing missing values (geom_point).

## Warning: Removed 330 rows containing missing values (geom_text).

# Player Points Predictions by Position

## Point Guard

final_pg_ppg_pts_predict <- pg %>%
  filter(PPG >= 10) %>% # This will filter out Point Guards who score under 10 Points Per Game
  mutate(pg_r_hat = predict(pg_overall_tidy_ppg, newdata = .)) %>%
  ggplot(aes(pg_r_hat, PTS, label = player_name)) +
  geom_point(colour = "Red", alpha = 0.8) +
  geom_text(nudge_x = 2, cex = 2) +
  geom_smooth() + # This formula will predict each Point Guards predicted Points Per Game against their actual output 
  labs(title = "Predicted in comparison to Actual Points Output",
       subtitle = "Identifying Point Guards who perform to expectated levels",
       x = "Predicted Points Per Game Average",
       y = "Total Points") + # Changing and configuring the Titles of the graph. 
  theme_linedraw() # Changing the theme of the graph.

## Shooting Guard

final_sg_ppg_pts_predict <- sg %>%
  filter(PPG >= 10) %>% # This will filter out Shooting Guards who score under 10 Points Per Game
  mutate(sg_r_hat = predict(sg_overall_tidy_ppg, newdata = .)) %>%
  ggplot(aes(sg_r_hat, PTS, label = player_name)) +
  geom_point(colour = "Red", alpha = 0.8) +
  geom_text(nudge_x = 2, cex = 2) +
  geom_smooth() + # This formula will predict each Shooting Guards predicted Points Per Game against their actual output
  labs(title = "Predicted in comparison to Actual Points Output",
       subtitle = "Identifying Shooting Guards who perform to expectated levels",
       x = "Predicted Points Per Game Average",
       y = "Total Points") + # Changing and configuring the Titles of the graph
  theme_linedraw() # Changing the theme of the graph.

## Small Forward

final_sf_ppg_pts_predict <- sf %>%
  filter(PPG >= 10) %>% # This will filter out Small Forwards who score under 10 Points Per Game
  mutate(sf_r_hat = predict(sf_overall_tidy_ppg, newdata = .)) %>%
  ggplot(aes(sf_r_hat, PTS, label = player_name)) +
  geom_point(colour = "Red", alpha = 0.8) +
  geom_text(nudge_x = 2, cex = 2) +
  geom_smooth() + # This formula will predict each Small Forwards predicted Points Per Game against their actual output
  labs(title = "Predicted in comparison to Actual Points Output",
       subtitle = "Identifying Small Forwards who perform to expectated levels",
       x = "Predicted Points Per Game Average",
       y = "Total Points") + # Changing and configuring the Titles of the graph
  theme_linedraw() # Changing the theme of the graph.

## Power Forward

final_pf_ppg_pts_predict <- pf %>%
  filter(PPG >= 10) %>% # This will filter out Power Forwards who score under 10 Points Per Game
  mutate(pf_r_hat = predict(pf_overall_tidy_ppg, newdata = .)) %>%
  ggplot(aes(pf_r_hat, PTS, label = player_name)) +
  geom_point(colour = "Red", alpha = 0.8) +
  geom_text(nudge_x = 2, cex = 2) +
  geom_smooth() + # This formula will predict each Power Forwards predicted Points Per Game against their actual output
  labs(title = "Predicted in comparison to Actual Points Output",
       subtitle = "Identifying Power Forwards who perform to expectated levels",
       x = "Predicted Points Per Game Average",
       y = "Total Points") + # Changing and configuring the Titles of the graph 
  theme_linedraw() # Changing the theme of the graph.

## Centre

final_c_ppg_pts_predict <- centres %>%
  filter(PPG >= 10) %>% # This will filter out Centres who score under 10 Points Per Game
  mutate(c_r_hat = predict(c_overall_tidy_ppg, newdata = .)) %>%
  ggplot(aes(c_r_hat, PTS, label = player_name)) +
  geom_point(colour = "Red", alpha = 0.8) +
  geom_text(nudge_x = 2, cex = 2) +
  geom_smooth() + # This formula will predict each Centres predicted Points Per Game against their actual output 
  labs(title = "Predicted in comparison to Actual Points Output",
       subtitle = "Identifying Centres who perform to expectated levels",
       x = "Predicted Points Per Game Average",
       y = "Total Points") + # Changing and configuring the Titles of the graph 
  theme_linedraw() # Changing the theme of the graph.

Final Team Selection

selected_team <- bind_rows(selected_player_c, selected_player_pf_2, selected_player_sf, selected_player_sg, selected_player_pg_2) # Combine above filtered players and combine to express the chosen side. 

datatable(selected_team, rownames = FALSE, filter="top", options = list(pageLength = 5, scrollX=T)) # The table describes the ideal team. Selection based on predicted value and current value based on output per game. Used the datatable funtion to include a search bar